M208 Pure Mathematics

Handbook

This publication forms part of an Open University course. Details of this and other Open University courses can be obtained from the Student Registration and Enquiry Service, The Open University, PO Box 197, Milton Keynes, MK7 6BJ, United Kingdom: tel. +44 (0)870 333 4340, e-mail general-enquiries@open.ac.uk

Alternatively, you may visit the Open University website at http://www.open.ac.uk where you can learn more about the wide range of courses and packs offered at all levels by The Open University.

To purchase a selection of Open University course materials, visit the webshop at www.ouw.co.uk, or contact Open University Worldwide, Michael Young Building, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom, for a brochure: tel. +44 (0)1908 858785, fax +44 (0)1908 858787, e-mail ouwenq@open.ac.uk

The Open University, Walton Hall, Milton Keynes, MK7 6AA.

First published 2006.

Copyright © 2006 The Open Universityc

All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, transmitted or utilised in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without written permission from the publisher or a licence from the Copyright Licensing Agency Ltd. Details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP.

Open University course materials may also be made available in electronic formats for use by students of the University. All rights, including copyright and related rights and database rights, in electronic course materials and their contents are owned by or licensed to The Open University, or otherwise used by The Open University as permitted by applicable law.

In using electronic course materials and their contents you agree that your use will be solely for the purposes of following an Open University course of study or otherwise as licensed by The Open University or its assigns.

Except as permitted above you undertake not to copy, store in any medium (including electronic storage or use in a website), distribute, transmit or re-transmit, broadcast, modify or show in public such electronic materials in whole or in part without the prior written consent of The Open University or in accordance with the Copyright, Designs and Patents Act 1988.

Edited, designed and typeset by The Open University, using the Open University TEX System.

Printed and bound in the United Kingdom by Hobbs the Printers Limited, Brunel Road, Totton, Hampshire SO40 3WX.

SUP 86903 5

1.1

This Handbook may be taken into the examination, and unrestricted handwritten annotation to it is allowed, but you are not permitted to add further sheets of notes, inserts or ‘post-its’.

Contents

Notation 5Greek alphabet 5Notation introduced in Introduction Block 5Notation introduced in Group Theory Block A 6Notation introduced in Linear Algebra Block 7Notation introduced in Analysis Block A 7Notation introduced in Group Theory Block B 8Notation introduced in Analysis Block B 8

Introduction Block 9Unit I1 Real functions and graphs 9Unit I2 Mathematical language 13Unit I3 Number systems 18

Group Theory Block A 23Unit GTA1 Symmetry 23Unit GTA2 Groups and subgroups 27Unit GTA3 Permutations 31Unit GTA4 Cosets and Lagrange’s Theorem 35

Linear Algebra Block 39Unit LA1 Vectors and conics 39Unit LA2 Linear equations and matrices 43Unit LA3 Vector spaces 48Unit LA4 Linear transformations 52Unit LA5 Eigenvectors 56

Analysis Block A 60Unit AA1 Numbers 60Unit AA2 Sequences 63Unit AA3 Series 66Unit AA4 Continuity 69

Group Theory Block B 73Unit GTB1 Conjugacy 73Unit GTB2 Homomorphisms 75Unit GTB3 Group actions 78

Analysis Block B 81Unit AB1 Limits 81Unit AB2 Differentiation 85Unit AB3 Integration 88Unit AB4 Power series 92

3

Appendix 95 Sketches of graphs of basic functions 95 Sketches of graphs of standard inverse functions 96 Properties of trigonometric and hyperbolic functions 97 Standard derivatives 98 Standard Taylor series 98 Standard primitives 99 Group tables of symmetry groups 100 Groups of small order 100 Three types of non-degenerate conic 101 Six types of non-degenerate quadric 101

Index 102

Wording of questions In the wording of TMA and Examination questions:write down or state means ‘write down without justification’;find, determine, calculate, explain, derive, evaluate or solve means ‘show allyour working’;prove, show or deduce means ‘justify each step’.In particular, if you use a definition, result or theorem to go from one lineto the next, then you must state clearly which fact it is that you are using.Also, remember that when you use a theorem, you must demonstrate thatall the conditions of the theorem are satisfied.

4

( )

Notation

Greek alphabet α A alpha ι I iota ρ P rho β B beta κ K kappa σ Σ sigma γ Γ gamma λ Λ lambda τ T tau δ ∆ delta µ M mu υ Υ upsilon ε E epsilon ν N nu φ Φ phi ζ Z zeta ξ Ξ xi χ X chi η H eta o O omicron ψ Ψ psi θ Θ theta π Π pi ω Ω omega

Notation introduced in Introduction Block (a, b) open interval, excluding endpoints a, b, x : a < x < b[a, b] closed interval, including endpoints a, b, x : a ≤ x ≤ b(a, b] half-open interval, excluding a, including b, x : a < x ≤ b[a, b) half-open interval, including a, excluding b, x : a ≤ x < b(−∞, a) open interval, x : x < a(−∞, a] closed interval, x : x ≤ a(a,∞) open interval, x : x > a[a,∞) closed interval, x : x ≥ a∞ infinityR set of real numbersR+ set of positive real numbers∗R set of non-zero real numbers

R2 set of points in the plane Q set of rational numbers Z set of integers, . . . ,−2,−1, 0, 1, 2, . . .N set of natural numbers, 1, 2, 3, . . .C set of complex numbers |x| modulus of number x [x] integer part of number x → tends to (for asymptotic behaviour and limits) f ′(x), f ′′(x) first and second derivatives of function f at x ∅ empty set a ∈ A a is an element of the set A A ⊆ B A is a subset of the set B A ⊂ B A is a proper subset of the set B A ∪ B union of sets A and B, x : x ∈ A or x ∈ BA ∩ B intersection of sets A and B, x : x ∈ A and x ∈ BA− B difference between sets A and B, x : x ∈ A, x /∈ Bn! n factorial, equal to n× (n− 1) × (n− 2) × · · · × 3 × 2 × 1

n n!binomial coefficient

k! (n− k)!, also denoted by nCkk

−→ maps to, for sets −→ maps to, for variables f−1 inverse of function f g f composite function with rule x −→ g(f(x)), where f and g are functions

5

P ⇒ Q if P , then Q (P implies Q)P ⇔ Q P if and only if Q (P is equivalent to Q)x + iy a complex number, where i2 = −1Re z real part of complex number zIm z imaginary part of complex number zz complex conjugate of complex number z|z| modulus of complex number zarg z an argument of complex number zArg z principal argument of complex number za ≡ b a is congruent to b (with respect to a particular modulus)Zn set of integers modulo n, 0, 1, . . . , n− 1a +n b remainder of a + b on division by na×n b remainder of a× b on division by nx ∼ y x is related to y (by a particular relation)[[x]] equivalence class of x (with respect to a particular equivalence relation)x · y alternative notation for x× y, rarely used in this course

Notation introduced in Group Theory Block A

S(F ) set of symmetries of plane figure F S+(F ) set of direct symmetries of plane figure F rθ rotation through θ (anticlockwise) about the centre of a disc (or the origin) qφ reflection in line through the centre of a disc (or the origin) at angle φ to the horizontal (G, ) set G with binary operation x−1 inverse of group element x, in multiplicative notation −x inverse of group element x, in additive notation e identity element of a group (or the constant e = 2.718 281 . . .) |G| order of group G |x| order of group element x R3 set of points in three-dimensional space 〈x〉 cyclic group generated by x ∼= is isomorphic to Cn typical cyclic group of order n, generated by x ∗Z set of non-zero integers modulo n, 1, 2, 3, . . . , n− 1n

Sn symmetric group of order n An alternating group of order n K4 Klein group (of order 4) gH,Hg left and right cosets of subgroup H in a particular group A . B composite, under set composition, of subsets A,B of a group (G, ), a b : a ∈ A, b ∈ BG/N quotient group of G by N nZ set of multiples of integer n Z+ set of positive integers

6

∑

Notation introduced in Linear Algebra Block ‖v‖ magnitude of vector v 0 zero vector (or zero matrix) i, j,k unit vectors in the directions of x-, y-, z-axes, respectively (a, b, c) a vector in R3 written in component form u . v dot product of vectors u and v v unit vector in the same direction as vector v e eccentricity of a conic (A | I) matrix A augmented by matrix I (aij ) matrix with (i, j)-entry aij

In identity matrix of size n× n AT transpose of matrix A A−1 inverse of matrix A det A determinant of matrix A Aij cofactor associated with entry aij of matrix A Rn set of all ordered n-tuples, called n-dimensional space Pn set of all real polynomials of degree less than n Mm,n set of all m× n matrices with real entries R∞ set of all infinite sequences of real numbers 〈S〉 span of finite set S of vectors vE E-coordinate representation of vector v (coordinates with respect to basis E) dim V dimension of vector space V iV identity linear transformation of vector space V Im(t) image of linear transformation t Ker(t) kernel of linear transformation t S(λ) eigenspace corresponding to eigenvalue λ

Notation introduced in Analysis Block A max E maximum element of set E min E minimum element of set E sup E supremum (least upper bound) of set E inf E infimum (greatest lower bound) of set E an sequence of numbers, a1, a2, . . . . lim an limit of sequence an as n tends to ∞

n→∞ ∞

an sum of series a1 + a2 + · · · n=1

sin−1 inverse of function sin (similarly for cos, tan, sinh, etc.) does not tend to

7

[ ]

( ) ( ) ( )

Notation introduced in Group Theory Block B

gHg−1 a conjugate subgroup of subgroup HM group of invertible 2 × 2 matrices under matrix multiplicationU group of invertible 2 × 2 upper-triangular matrices under matrix multiplicationL group of invertible 2 × 2 lower-triangular matrices under matrix multiplicationV group of 2 × 2 matrices with determinant 1 under matrix multiplicationIm(φ) image of homomorphism φKer(φ) kernel of homomorphism φg ∧ x set element obtained when group element g acts on set element xOrb(x) orbit of xStab(x) stabiliser of xFix(g) fixed set (or fixed point set) of g←→ corresponds to∗C set of non-zero complex numbers

Notation introduced in Analysis Block B Nr (c) punctured neighbourhood, (c − r, c) ∪ (c, c + r), r > 0lim limit as x tends to cx→c

lim limit as x tends to c from the right x→c+

lim limit as x tends to c from the left −x→c

dy Leibniz notation for derivative of y = f(x) with respect to x

dxf (n) nth derivative of function ff ′

L(c) left derivative of f at cf ′

R(c) right derivative of f at cInt(I) interior of interval I, i.e. largest open subinterval of IP partition of an interval‖P ‖ mesh of partition P , length of longest subintervalmi inff(x) : xi−1 ≤ x ≤ xi, where f is a function on [xi−1, xi]Mi supf(x) : xi−1 ≤ x ≤ xi, where f is a function on [xi−1, xi]L(f, P ) lower Riemann sum of function f over partition PU(f, P ) upper Riemann sum of function f over partition P∫ b

f lower integral of function f over interval [a, b] a−∫−b

f upper integral of function f over interval [a, b] a∫ b

f integral of function f over interval [a, b] ∫a

f(x) dx a primitive of function f

aF (x)

b primitive F evaluated between a and b, F (b) − F (a)∼ f(n) ∼ g(n) as n →∞ means f(n)/g(n) → 1 as n →∞ Tn(x) Taylor polynomial of degree nRn(x) remainder term for Taylor polynomial of degree nR radius of convergence of a power series

α generalised binomial coefficient, α = 1, α =

α(α − 1)(α − 2) · · · (α − n + 1), n ∈ N

n 0 n n!

8

Introduction Block

I1 Real functions and graphs

1 Real functions

1 A real function f is defined by specifying

• a set of real numbers A, called the domain of f ; • a set of real numbers B, called the codomain

of f ; • a rule that associates with each real number x in

the set A a unique real number f (x) in the set B.

The number f (x) is the image of x under f or the value of f at x. Convention When a real function is specified only by a rule, it is to be understood that the domain of the function is the set of all real numbers for which the rule is applicable, and the codomain of the function is R.

2 Intervals are denoted as follows. In the diagrams, an open circle indicates that an endpoint is excluded, and a solid circle • indicates that an endpoint is included. open intervals

(a, b) (a, ∞) c c c a b a

a < x < b x > a

(−∞, b) (−∞, ∞) c b

x < b R

closed intervals

[a, b] [a, ∞) s s s a b a a ≤ x ≤ b x ≥ a

(−∞, b] (−∞, ∞) s b x ≤ b R

half-open (or half-closed) intervals

[a, b) (a, b] s c c s a b a b a ≤ x < b a < x ≤ b

I1

3 A linear function has rule of the form

f (x) = ax + b, where a = 0. A quadratic function has rule of the form

f (x) = ax 2 + bx + c, where a = 0. In completed-square form this is written as

f (x) = a(x − α)2 + β,

where b 4ac − b2

α = − , β = .2a 4a

The graph of this function is a parabola with vertex at (α, β) and axis x = α. A cubic function has rule of the form

f (x) = ax 3 + bx2 + cx + d, where a = 0. The graph of a cubic function crosses the x-axiseither once or three times.A linear rational function has rule of the form

ax + b f (x) =

cx + d, where ad − bc = 0.

The graph of a linear rational function is a hyperbola lying between the asymptotes y = a/c and x = −d/c. The reciprocal function has rule

f (x) = 1/x, for x = 0. An exponential function has rule of the form

xf (x) = a , where a > 0. The modulus function has rule

f (x) = |x| = x, x ≥ 0,

−x, x ≤ 0. The graph of the modulus function has a ‘corner’ atthe origin.The integer part function has rule

f (x) = [x] = largest integer less than or equal to x.

The graph of the integer part function has a ‘jump’ at each integer value of x.

4 The translation (x, y) −→ (x + α, y + β) applied to the graph with equation y = f (x) gives the graph with equation

y − β = f (x − α); that is, y = f (x − α) + β.

The graph is shifted by α units to the right, and by βunits upwards. (0, 0) is shifted to (α, β).The scaling (x, y) −→ (λx, µy) applied to the graph with equation y = f (x) gives the graph with equation

y/µ = f (x/λ); that is, y = µf (x/λ). The graph is scaled by λ in the x-direction, and by µ in the y-direction. (1, 1) is shifted to (λ, µ), and (0, 0) remains fixed.

9

I1

2 Graph sketching

1 Determining features of a graph

Domain

The domain of a real function specified just by a rule is the set of all real numbers, excluding any numbers which give an expression which is not defined—for example, a zero in the denominator of a rational function, or the square root of a negative number.

Symmetry features

A graph may possess certain types of symmetry. • For a periodic function, such as a

trigonometric function, the graph is unchanged by a translation along the x-axis through the period, p say:

f(x + p) = f(x). • For an even function, the graph is unchanged

by reflection in the y-axis:f(−x) = f(x).

• For an odd function, the graph is unchanged by rotation through an angle π about the origin:

f(−x) = −f(x).

Intercepts

An intercept is a value of x or y at which the graph y = f(x) of a function f meets the x- or y-axis, respectively. The x-intercepts are the solutions (if any) of the equation f(x) = 0 and are also known as the zeros of f . The y-intercept is the value f(0), if this exists.

Intervals on which a function has constant sign

Let f be a real function with domain A. Then:f is positive on an interval I in A if f(x) > 0for all x in I;f is negative on an interval I in A if f(x) < 0for all x in I;f has a zero at the point a in A if f(a) = 0.

Sometimes we can find the intervals on which afunction has constant sign by using a sign table.

Intervals on which a function is increasing/decreasing

A function f is increasing on an interval I if for all x1, x2 ∈ I,

if x1 < x2, then f(x1) ≤ f(x2). A function f is strictly increasing on an interval I if for all x1, x2 ∈ I,

if x1 < x2, then f(x1) < f(x2). A function f is decreasing on an interval I if for all x1, x2 ∈ I,

if x1 < x2, then f(x1) ≥ f(x2). A function f is strictly decreasing on an interval I if for all x1, x2 ∈ I,

if x1 < x2, then f(x1) > f(x2).

Increasing/decreasing criterion

1. If f ′(x) > 0 for all x in an interval I, then f is (strictly) increasing on I.

2. If f ′(x) < 0 for all x in an interval I, then f is (strictly) decreasing on I.

A stationary point of f is a value a such that the tangent to the graph of f is horizontal at the point (a, f(a)).

First Derivative Test Suppose that a is a stationary point of a differentiable function f ; that is, f ′(a) = 0.

• If f ′(x) changes from positive to negative as x increases through a, then f has a local maximum at a.

• If f ′(x) changes from negative to positive as x increases through a, then f has a local minimum at a.

• If f ′(x) remains positive or remains negative as x increases through a (except at a itself, where f ′(a) = 0), then f has a horizontal point of inflection at a.

2 Asymptotic behaviour

An asymptote for the graph of a function is a straight line which is approached more and more closely by the graph when the domain variable x or the codomain variable y (or both) takes very large (positive or negative) values.

An asymptote with an equation of the form x = a isa vertical asymptote.An asymptote with an equation of the form y = b is ahorizontal asymptote.The behaviour of a function f near a verticalasymptote x = a may take various forms. Wedescribe the behaviour shown in the above diagramnear the vertical asymptote x = a as follows.

10

f (x) takes arbitrarily large values as x tends to a from the right;

we write this in symbols as +f (x) →∞ as x → a ,

and read it as

f (x) tends to infinity as x tends to a from the right.

Similarly, f (x) takes arbitrarily large negative values as x tends to a from the left;

we write this in symbols as −f (x) → −∞ as x → a ,

and read it as

f (x) tends to minus infinity as x tends to a from the left.

3 Polynomial and rational functions

In general, the behaviour of a polynomial function of degree n,

f (x) = anx n + an−1x n−1 + · · ·+ a1x + a0, where an = 0,

for large values of x, is similar to that of the term anxn . We call xn the dominant term. This behaviour is summarised in the following tables.

an > 0 x →∞ x → −∞

n even f (x) →∞ f (x) →∞

n odd f (x) →∞ f (x) → −∞

an < 0 x →∞ x → −∞

n even f (x) → −∞ f (x) → −∞

n odd f (x) → −∞ f (x) →∞

A rational function is a function defined by a rule of the form

p(x) x −→

q(x) ,

where both p and q are polynomial functions. Locating vertical and horizontal asymptotes is an important step in sketching the graph of any rational function. Vertical asymptotes occur at the values of x for which q(x) = 0 and p(x) = 0, and horizontal asymptotes may occur when x →∞ or x → −∞. To find the behaviour of a rational function for large values of x, we divide both the numerator and the denominator by the dominant term of the denominator and consider the value of f (x) as x → ±∞.

I1

Strategy 2.1 Graph-sketching strategy To sketch the graph of a given function f , determine the following features of f (where possible), and show these features in your sketch. 1. The domain of f . 2. Whether f is even, odd or periodic (or none

of these). 3. The x-intercepts and y-intercept of f . 4. The intervals on which f is positive or

negative.5. The intervals on which f is increasing or

decreasing, the nature of any stationary points, and the value of f at each of these points.

6. The asymptotic behaviour of f .

For some graphs, we can obtain sufficient information from only some of the steps.

4 There is an alternative test for a local maximum or local minimum, using the second derivative of the function f .

Second Derivative Test Suppose that a is a stationary point of a differentiable function f ; that is, f ′(a) = 0.

1. If f ′′(a) < 0, then f has a local maximum at a.

2. If f ′′(a) > 0, then f has a local minimumat a.

3 New graphs from old

1 Further graph-sketching techniques

To sketch the graph of a combination of two functions, one of which is a trigonometric function, it is often convenient to use other simple graphs as ‘construction lines’, and to exploit known properties of the sine and cosine functions.

Strategy 3.1Extended graph-sketching strategyTo sketch the graph of a given function f , determine the features of f listed in steps 1–6 of Strategy 2.1, and the following features. 7. Any appropriate construction lines, and the

points where f meets these lines.

11

I1

2 A composite function is a function obtained by applying first one function and then another. A hybrid function is defined by different formulas on different parts of its domain.

4 Hyperbolic functions

1 Properties of the exponential function

The function exp with rule f(x) = ex has the following properties. 1. The domain of exp is R. 2. exp is not even, odd or periodic. 3. ex > 0 for all x in R, so exp is positive on R.

x4. exp is its own derivative; that is, if f(x) = e , xthen f ′(x) = e .

Since ex > 0 for all x in R, exp is increasing on R.5. e0 = 1, ex > 1 for all x > 0, and ex < 1 for all

x+yx < 0; e = ex × ey for all x, y in R. 6. For each positive integer n, ex/xn →∞ as

x →∞. (We sometimes express this property by saying that ex grows faster than any polynomial when x is large.)

7. ex →∞ as x →∞ and ex → 0 as x → −∞.

2 Hyperbolic functions

• cosh is the hyperbolic cosine function, with rule

ex + e−x

cosh x = .2

• sinh is the hyperbolic sine function, with rule

ex − e−x

sinh x = .2

• tanh is the hyperbolic tangent function,

5 Curves from parameters

1 Parametric equations for a curve have the form

x = f(t), y = g(t), where f and g are real functions of the parameter t. Both f and g have the same domain, which is usually an interval I of the real line. The corresponding parametrisation is the function

α(t) = (f(t), g(t)), for t in I.

A parametrisation of a given curve need not be unique. Different parametrisations of a curve may correspond to different modes of traversing the curve.

2 A cycloid is the path traced out by a point on the edge of a wheel as the wheel rolls along a horizontal surface without slipping. The standard parametrisation of a cycloid corresponding to a wheel of unit radius is

x = t − sin t, y = 1 − cos t, for t in R.

3 Below, we summarise the standard parametrisations for lines, conics and two other curves.

Line through (p, q) and (r, s): s − q

y − q = (x − p), r − p

α(t) = (p + (r − p)t, q + (s − q)t), for t in R.

Circle with centre (0, 0), radius a: 2 2 x 2 + y = a ,

with rulesinh x

tanh x = .cosh x

• sech is the hyperbolic secant function, with rule

1sech x =

cosh x.

• cosech is the hyperbolic cosecant function, with rule

1cosech x = .

sinh x • coth is the hyperbolic cotangent function,

with rule1

coth x = .tanh x

Some properties of the trigonometric and hyperbolic functions are given in a table on page 97.

α(t) = (a cos t, a sin t),

Ellipse in standard form: 2 2x

+ y

= 1, a2 b2

α(t) = (a cos t, b sin t),

for t in [0, 2π].

for t in [0, 2π].

12

I2

Parabola in standard form: I2 Mathematical language y 2 = 4ax,

2α(t) = (at , 2at), for t in R.

Hyperbola in standard form: 2 2x y− = 1,

a2 b2

α(t) = (a sec t, b tan t), for t in [−π, π], excluding −π/2 and π/2,

or α(t) = (a cosh t, b sinh t), for t in R (right-hand part only).

1 What is a set?

1 A set is a collection of objects, such as numbers, points, functions, or even other sets. Each object in a set is an element or member of the set, and the elements belong to the set, or are in the set. We can illustrate a set S by a diagram called a Venn diagram, as in the example below.

2 Sets of numbers

Cardioid in standard form:α(t) = (2 cos t + cos 2t, 2 sin t + sin 2t),for t in [−π, π].

R is the set of real numbers. R ∗ is the set of non-zero real numbers. Q is the set of rational numbers. Z is the set of integers . . . , −2, −1, 0, 1, 2, . . . . N is the set of natural numbers 1, 2, 3, . . . . A prime number is an integer n, greater than 1,whose only positive factors are 1 and n; the first few primes are 2, 3, 5, 7, 11, 13, 17.

3 To indicate that a is an element of the set A, we write a ∈ A.To indicate that b is not an element of the set A, we

Trisectrix in standard form: α(t) = (cos t + cos 2t, sin t + sin 2t), for t in [−π, π].

write b /∈ A.

4 A set with only one element, such as the set 2, is called a singleton.The empty set has no elements, and is denotedby ∅.

5 The solution set of an equation, or a system ofequations, is the set of its solutions. It depends onthe set from which the solutions are taken.

6 Plane sets

A set of points in R2 is called a plane set or aplane figure. Simple examples of plane sets arelines and circles.A straight line l with slope a and y-intercept b iswritten as

l = (x, y) ∈ R2 : y = ax + b. We sometimes refer to ‘the line y = ax + b’ as a shorthand way of specifying this set.The set of points on one side of a line, possiblytogether with all the points on the line itself, is known as a half-plane. The unit circle U is written as (x, y) ∈ R2 : x 2 + y 2 = 1.

13

( )

( ) ( ) ( )

( ) ( )

I2

A circle C of radius r centred at the point (a, b) is written as

2C = (x, y) ∈ R2 : (x − a)2 + (y − b)2 = r . The set of points inside a circle, possibly together with all the points on the circle, is known as a disc. In a diagram of a plane set, when the set illustrated does not include a boundary line, we denote the boundary by a broken line.

7 The graph of a real function f : A −→ R is the set (x, f (x)) : x ∈ A .

8 Two sets A and B are equal if they have exactly the same elements; we write A = B. A set A is a subset of a set B if each element of A is also an element of B. We also say that A is contained in B, and we write A ⊆ B. We sometimes indicate that a set A is a subset of a set B by reversing the symbol ⊆ and writing B ⊇ A, which we read as ‘B contains A’. To indicate that A is not a subset of B, we write A B. We may also write this as B A, which we read as ‘B does not contain A’. If a set A is a subset of a set B that is not equal to B, then we say that A is a proper subset of B, and we write A ⊂ B or B ⊃ A.

Strategy 1.1 To show that two sets A and B are equal:

show that A ⊆ B; show that B ⊆ A.

9 A finite set is a set which has a finite number of elements; that is, the number of elements is some natural number, or 0. A set with n elements has 2n subsets.

10 For any positive integer n, we define n! (read as ‘n factorial’) by

n! = n × (n − 1) × (n − 2) × · · · × 3 × 2 × 1. Also, 0! = 1. For any non-negative integers n and k with k ≤ n,

n n! = .

k k! (n − k)! This expression is called a binomial coefficient. It is the number of subsets with k elements of a setwith n elements.If n and k are positive integers with 1 ≤ k ≤ n, then

n n n + 1 = k − 1 +

k k.

If n and k are positive integers with 0 ≤ k ≤ n, then

n n = . n − k k

14

11 Set operations

Let A and B be any two sets. The union of A and B is the set

A ∪ B = x : x ∈ A or x ∈ B .

The intersection of A and B is the set A ∩ B = x : x ∈ A and x ∈ B .

Two sets with no element in common are disjoint. The difference between A and B is the set

A − B = x : x ∈ A, x /∈ B .

Note that A − B is different from B − A, when A = B.

2 Functions

1 Functions

A function f is defined by specifying: • a set A, called the domain of f ; • a set B, called the codomain of f ; • a rule x −→ f (x) that associates with each

element x ∈ A a unique element f (x) ∈ B.

The element f (x) is the image of x under f . Symbolically, we write

f : A −→ Bx −→ f (x).

We often refer to a function as a mapping, and say that f maps A to B and x to f (x).

2 A function of the form f : I −→ R2, where I is aninterval of R, can be used to parametrise a curve in the plane.

3 The identity function on a set A is the function

iA : A −→ A x −→ x.

4 Given a function f : A −→ B and a subset S of A, the image, or image set, of S under f , written f (S), is the set

f (S) = f (x) : x ∈ S . The image, or image set, of the function f is the image of its whole domain,

f (A) = f (x) : x ∈ A .

5 A function f : A −→ B is onto if f (A) = B.

6 A function f : A −→ B is one-one if each element of f (A) is the image of exactly one element of A; that is,

if x1, x2 ∈ A and f (x1) = f (x2), then x1 = x2.

A function that is not one-one is many-one.

7 Inverse functions

Let f : A −→ B be a one-one function. Then f has an inverse function f −1 : f (A) −→ A, with rule

f −1(y) = x, where y = f (x).

A function f : A −→ B that is both one-one and onto has an inverse function f −1 : B −→ A. Such a function f is said to be a one-one correspondence, or bijection, between the sets A and B. Sometimes we can obtain an inverse function for a function which is not one-one by restricting the domain. Let f : A −→ B and let C be a subset of the domain A. Then the function g : C −→ B defined by

g(x) = f (x), for x ∈ C,

is the restriction of f to C.

8 Composite functions

Let f : A −→ B and g : C −→ D be any two functions; then the composite function g f has

domain x ∈ A : f (x) ∈ C , codomain D, rule (g f )(x) = g(f (x)).

This definition allows us to consider the composite of any two functions, although in some cases the domain may turn out to be the empty set ∅.

I2

Strategy 2.1 To show that the function g : B −→ A is the inverse function of thefunction f : A −→ B.1. Show that f (g(x)) = x for each x ∈ B; that

is, f g = iB . 2. Show that g(f (x)) = x for each x ∈ A; that

is, g f = iA.

3 The language of proof

1 Mathematical statements

A mathematical statement (sometimes called a proposition) is an assertion that is either true or false, although we may not know which. Every statement has a related statement, called its negation, which is true when the original statement is false, and false when the original statement is true. A theorem is a true mathematical statement (usually important). A lemma is a less important theorem that is useful when proving other theorems. A corollary is a theorem that follows from another theorem by a short additional argument.

2 Implications

An implication is a statement of the form

if P, then Q,

where P is a statement, called the hypothesis, and Q is a statement, called the conclusion.Ways of writing the implication ‘if P , then Q’:

P implies QP ⇒ QQ whenever PQ follows from PP is sufficient for QQ is necessary for PP only if Q

The converse of the implication ‘if P , then Q’ is the implication ‘if Q, then P ’.

3 Equivalences

The statement ‘if P , then Q, and if Q, then P ’ is usually expressed more concisely as

P if and only if Q.

Such a statement is called an equivalence. Ways of writing ‘P if and only if Q’:

P ⇔ Q P is equivalent to Q P is necessary and sufficient for Q

15

I2

4 A proof of a mathematical statement is a logical argument that establishes that the statement is true.

5 Proof by exhaustion If there are only a small number of possibilities to consider, we may be able to prove a statement by considering each possibility in turn.

6 Direct proof In general, to prove that the implication P ⇒ Q is true, we start by assuming that P is true, and build up a sequence of statements P, P1, P2, . . . , Q, each of which follows from one or more statements further back in the sequence or from previous mathematical knowledge. To prove that the equivalence P ⇔ Q is true, we have to prove both of the implications P ⇒ Q and Q ⇒ P . A statement Q that is not an implication can be proved in a similar way to an implication P ⇒ Q, by constructing a sequence of statements as above, but the initial statement P will be not an assumption but instead a statement that we know to be true from our previous knowledge.

7 To prove that an implication P ⇒ Q is false, we give one example of a case where the statement P is true but the statement Q is false. Such an example is called a counter-example to the implication.

8 Proof by induction

Principle of Mathematical Induction To prove that a statement P (n) is true for n = 1, 2, . . . . 1. Show that P (1) is true. 2. Show that the implication P (k) ⇒ P (k + 1)

is true for k = 1, 2, . . . .

The Principle of Mathematical Induction can be adapted to prove that a statement P (n) is true for all integers n greater than or equal to some given integer other than 1.

9 Proof by contradiction To prove that a statement Q is true, begin by assuming that Q is false. Then attempt to deduce, using a sequence of statements, a statement that is definitely false, which in this context is called a contradiction. If this can be achieved, then we can conclude that the assumption is false—in other words, Q is true.

10 To prove an implication P ⇒ Q using proof by contradiction, begin by assuming that P is true and Q is false, and deduce a contradiction.

11 Proof by contraposition Given any implication, we can form another implication, called its contrapositive, which is equivalent to the original implication. The contrapositive of the implication ‘if P , then Q’ is ‘if not Q, then not P ’, where ‘not P ’ and ‘not Q’ denote the negations of the statements P and Q, respectively. Sometimes the easiest way to prove an implication is to prove its contrapositive instead.

12 Proof by splitting into cases Sometimes it can be helpful to consider different cases separately. For example, to prove a statement for all integers n, we could consider the cases n < 0, n = 0 and n > 0 separately.

13 Fundamental Theorem of Arithmetic Every integer greater than 1 has a unique expression as a product of primes.

14 Some properties of numbers

For n = 1, 2, . . . , 21 + 3 + · · ·+ (2n− 1) = n ,

1 + 2 + · · ·+ n = 1 n(n + 1),2 212 + 22 + · · ·+ n = 1 n(n + 1)(2n + 1),6 3 113 + 23 + · · ·+ n = n 2(n + 1)2 .4

For any real number x, and n = 1, 2, . . . ,

x n − 1 = (x− 1)(x n−1 + x n−2 + · · ·+ x + 1). There are infinitely many prime numbers. If an integer n > 1 is not divisible by any of the √primes less than or equal to n, then n is a prime number.

15 Statements of the type

x 2 ≥ 0 for all real numbers x. Every multiple of 6 is divisible by 3.

21 + 3 + 5 + · · ·+ (2n− 1) = n for each n ∈ N. Any rational number is a real number.

are known as universal statements, and the phrase ‘for all’ (and its equivalents) is referred to as the universal quantifier, sometimes denoted by the symbol ∀. 16 Statements of the type

There exists a real number . . . There is a real number x such that . . . Some multiples of 3 are not divisible by 6.

3The equation x = c has at least one real solution. are known as existential statements, and the phrase ‘there exists’ (and its equivalents) is referred to as the existential quantifier, sometimes denoted by the symbol ∃. 17 The negation of a universal statement is an existential statement, and vice-versa.

16

( ) ( )

4 Two identities

1 An identity is an equation involving variables which is true for all possible values of the variables.

2 The coefficients that appear in the expansions of (a + b)n, for n = 1, 2, . . . , can be arranged as a triangular table, in which 1s appear on the left and right edges, and the remaining entries can be generated by using the rule that each inner entry is the sum of the two nearest entries in the row above. This table is known as Pascal’s triangle. The first few rows are shown below.

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1 1 5 10 10 5 1

Theorem 4.1 Binomial Theorem Let a, b ∈ R and let n be a positive integer. Then ( ) ( )

n(a + b)n = n

a n + 1 a n−1b + · · · 0

n bn+ a n−kbk + · · ·+

n.

k n

3 Geometric series

Theorem 4.2 Geometric Series Identity Let a, b ∈ R and let n be a positive integer. Then

n a n − b = (a− b)(a n−1 + a n−2b + · · · + abn−2 + bn−1).

CorollarySum of a finite geometric seriesLet a, r ∈ R and let n be a positive integer. Then

n−1 a + ar + ar 2 + · · ·+ ar⎧ ( ) n⎨ 1 − r

a , if r = 1,= 1 − r⎩ na, if r = 1.

I2

4 Factorising polynomials

A polynomial in x of degree n is an expression of the form

anx n + an−1x n−1 + · · ·+ a1x + a0,

where an = 0.

Theorem 4.3 Polynomial Factorisation Theorem Let p be a polynomial of degree n and let α ∈ R. Then p(α) = 0 if and only if

p(x) = (x− α)q(x), where q is a polynomial of degree n− 1.

Corollary Let p(x) = x n + an−1x n−1 + · · ·+ a1x + a0, (∗)

and suppose that p(x) has n distinct real roots, α1, α2, . . . , αn. Then

p(x) = (x− α1)(x− α2) · · · (x− αn).

It can be shown that every polynomial of the form (∗) has a factorisation of the above form, although the roots need not be distinct and may include non-real complex numbers. Two useful consequences of this factorisation are

an−1 = −(α1 + α2 + · · ·+ αn) and

a0 = (−1)nα1α2 · · ·αn.

17

I3

I3 Number systems

1 Real numbers

1 Real numbers and rational numbers

Each real number can be represented as a point on a number line, known as the real line. Conversely, each point on the real line represents a real number. Each real number can be expressed as a (possibly infinite) decimal, such as 1.75 or − 0.333 . . . , and each such decimal expresses a real number. The rational numbers are the real numbers that can be expressed as fractions. Rational numbers have decimals that are either finite or recurring (have a repeating pattern of digits). The real numbers that are not rational are called irrational numbers. These numbers have decimals √ with no repeating patterns. The numbers 2, π and e are irrational. The set of real numbers and the set of rational numbers are denoted by R and Q, respectively. Thus Q ⊆ R.

2 Arithmetic in R

Addition A1 If a, b ∈ R, then a + b ∈ R. closure A2 If a ∈ R, then

a + 0 = 0 + a = a. identity A3 If a ∈ R, then there is a number

− a ∈ R such that a + (− a) = (− a) + a = 0. inverses

A4 If a, b, c ∈ R, then (a + b) + c = a + (b + c). associativity

A5 If a, b ∈ R, then a + b = b + a. commutativity

Multiplication M1 If a, b ∈ R, then a × b ∈ R. closure M2 If a ∈ R, then

a × 1 = 1 × a = a. identity M3 If a ∈ R − 0 , then there is a

number a−1 ∈ R such that a × a−1 = a−1 × a = 1. inverses

M4 If a, b, c ∈ R, then (a × b) × c = a × (b × c). associativity

M5 If a, b ∈ R, then a × b = b × a. commutativity

Addition and multiplication D If a, b, c ∈ R, then

a × (b + c) = a × b + a × c. distributivity

−1In properties A3 and M3, the numbers − a and aare known as the additive inverse (or negative) of a and the multiplicative inverse (or reciprocal) of a, respectively. The rational numbers Q also satisfy all the above properties; that is, the properties still hold if R is replaced by Q throughout. A set satisfying all these properties is known as a field.

3 A polynomial equation in x of degree n is an equation of the form p(x) = 0, where p(x) is a polynomial of degree n. Polynomial equations (and polynomials) of degrees 1, 2 and 3 are called linear, quadratic and cubic, respectively.

2 Complex numbers

1 Complex numbers

A complex number is an expression of the form x + iy, where x and y are real numbers and i2 = − 1.The set of all complex numbers is denoted by C.A complex number z = x + iy has real part x andimaginary part y; we write

Re z = x and Im z = y.

Two complex numbers are equal when their realparts and their imaginary parts are equal.The zero complex number, 0 + 0i, is written as 0.

A complex number of the form 0 + iy (where y = 0) is sometimes called an imaginary number.

2 The complex plane

There is a one-one correspondence between the complex numbers and the points in the plane, given by

f : C −→ R2

x + iy −→ (x, y). When we represent complex numbers by points in the plane, we refer to the plane as the complex plane, and we often refer to the complex numbers as points in the complex plane. Such a representation iscalled an Argand diagram.Real numbers are represented in the complex planeby points on the x-axis; this axis is called the realaxis. Similarly, numbers of the form iy arerepresented by points on the y-axis; this axis is calledthe imaginary axis.

18

√

6

I3

7 Arithmetic in C

The set of complex numbers C satisfies all theproperties previously given for arithmetic in R.(See page 18.)In particular, 0 = 0 + 0i plays the same role in C asthe real number 0 does in R, and 1 = 1 + 0i plays thesame role as 1. These numbers are called theidentities for addition and multiplication,respectively.The additive inverse (or negative) of z = x + iy is −z = −x − iy, and the multiplicative inverse (or reciprocal) of z = x + iy is

z x − iy= for z = 0. |z|2 x2 + y2

,

Unlike the real numbers, the complex numbers are not ordered.

8 Polar form

A non-zero complex number z = x + iy is in polar form if it is expressed as

r(cos θ + i sin θ), where r = |z| and θ is any angle (measured in radians anticlockwise) between the positive direction of the x-axis and the line joining z to the origin. Such an angle θ is called an argument of the complex number z, and is denoted by arg z. The principal argument of z is the value of arg z that lies in the interval (−π, π], and is denoted by Arg z. Sometimes we refer to z = x + iy as the Cartesian form of z, to distinguish it from the polar form.

9 Converting from and to polar form

The values in the table below will help you in some special cases.

θ 0 π/6 π/4 π/3 π/2 √

1 1√ 3sin θ 0 1

3 Complex arithmetic

Arithmetical operations on complex numbers are carried out as for real numbers, except that we replace i2 by −1 wherever it occurs. Let z1 = x1 + iy1 and z2 = x2 + iy2 be any complex numbers. Then the following operations can be applied.

Addition z1 + z2 = (x1 + x2) + i(y1 + y2) Subtraction z1 − z2 = (x1 − x2) + i(y1 − y2) Multiplication z1z2 = (x1x2 − y1y2) + i(x1y2 + y1x2)

4 Complex conjugates

The zcomplex conjugate of the complex number z = x + iy is the complex number x − iy. Properties of complex conjugates Let z1, z2

and z be any complex numbers. Then: 1. z1 + z2 = z1 + z2; 2. z1z2 = z1 × z2; 3. z + z z;= 2 Re

4. z − z = 2i Im z.

5 The modulus of a complex number

The modulus |z| of a complex number z is the distance from the point z in the complex plane to theorigin.Thus the modulus of the complex number z = x + iyis

2|z| = x2 + y .

Properties of modulus

1. |z| ≥ 0 for any z ∈ C, with equality only when z = 0.

2. |z1z2| = |z1| |z2| for any z1, z2 ∈ C.

Distance Formula The distance between the points z1 and z2 in the complex plane is |z1 − z2|. Conjugate–modulus properties

2 2 21. z = |z for all z ∈ C.| | ||2

√ 3 1√ 1cos θ 1 02. for all z ∈ C.|zz = 2 2 2z

Division of complex numbers The following formulas are also helpful:The second of the conjugate–modulus properties enables us to find reciprocals of complex numbers and to divide one complex number by another. As for real numbers, we cannot find a reciprocal of zero, nor divide any complex number by zero.

1 1 × z zReciprocal = =

2 , for z = 0.

z z × z |z|Quotient

z1 = z1 × z2 =

z1z2 for z2 = 0. z2 z2 × z2 |z2|2

,

sin(π − θ) = sin θ, cos(π − θ) = − cos θ; sin(−θ) = − sin θ, cos(−θ) = cos θ.

To convert a complex number from polar form to Cartesian form, use the equations

x = r cos θ, y = r sin θ.

(Conversion from Cartesian form to polar form follows.)

19

√

I3

To convert a non-zero complex number z from Cartesian form x + iy to polar form r(cos θ + i sin θ), first find the modulus r using the formula

2r = x2 + y .

If z is either real or imaginary, then it lies on one of the axes and has principal argument 0, π/2, π or −π/2. Otherwise, to find the principal argument θ, first find the first-quadrant angle φ that satisfies the equation

|x|cos φ = .

r Then determine the quadrant in which z lies (from the values of x and y; it may be helpful to sketch z on an Argand diagram), and calculate θ from φ by using the appropriate equation from the diagram below.

10 If z1 = r1(cos θ1 + i sin θ1), z2 = r2(cos θ2 + i sin θ2),

then

z1z2 = r1r2(cos(θ1 + θ2) + i sin(θ1 + θ2)) and

z1 r1 = (cos(θ1 − θ2) + i sin(θ1 − θ2)),z2 r2

provided that z2 = 0.

In particular, if z = r(cos θ + i sin θ) with r = 0, then the reciprocal of z is

1 1 = (cos(−θ) + i sin(−θ)).

z r

Strategy 2.1/2.2 To multiply two or more complex numbers given in polar form, multiply their moduli and add their arguments. To divide a complex number z1 by a non-zero complex number z2 when both are given in polar form, divide the modulus of z1 by the modulus of z2, and subtract the argument of z2

from the argument of z1.

Theorem 2.1 de Moivre’s Theorem If z = cos θ + i sin θ, then, for any n ∈ Z,

n z = (cos θ + i sin θ)n = cos nθ + i sin nθ.

20

11 Roots of a complex number

If a is a complex number, then the solutions of the equation zn = a are called the nth roots of a. Let a = ρ(cos φ + i sin φ) be a complex number in

npolar form. Then, for any n ∈ N, the equation z = a has n solutions, given by ( ( ) ( ))

φ 2kπ φ 2kπ = ρ1/nz cos + + i sin + ,

n n n n for k = 0, 1, . . . , n− 1. The nth roots of a complex number are equally spaced around a circle with centre the origin. The nth roots of 1 are known as the nth roots of unity.

12 Roots of polynomials

If p(z) is a polynomial, then the solutions of the polynomial equation p(z) = 0 are called the roots (or zeros) of p(z). (Thus the nth roots of the complex number a are the roots of the polynomial zn − a.) Every polynomial equation with complex coefficients has a complex solution; this is the Fundamental Theorem of Algebra.

Theorem 2.2Polynomial Factorisation TheoremLet p(z) be a polynomial of degree n with coefficients in C and let α ∈ C. Then p(α) = 0 if and only if

p(z) = (z − α)q(z), where q(z) is a polynomial of degree n− 1 with coefficients in C.

Corollary Every polynomial p(z) = anzn + an−1z

n−1 + · · ·+ a1z + a0, where n ≥ 1, ai ∈ C for each i and an = 0, has a factorisation

p(z) = an(z − α1)(z − α2) · · · (z − αn), where the complex numbers α1, α2, . . . , αn are the roots (not necessarily distinct) of p(z).

13 The complex exponential function

If z = x + iy, then z e = e x eiy = e x(cos y + i sin y).

Euler’s formula If y ∈ R, then

eiy = cos y + i sin y.

iθ isA complex number expressed in the form z = resaid to be in exponential form.Using this notation, de Moivre’s Theorembecomes the simple result

iθ)n inθ(e = e , for all θ ∈ R and all n ∈ Z.

3 Modular arithmetic

1 The Division Algorithm describes the result of dividing an integer a by a positive integer n.

Theorem 3.1 Division Algorithm Let a and n be integers, with n > 0. Then there are unique integers q and r such that

a = qn + r, with 0 ≤ r < n.

We say that dividing a by the divisor n gives quotient q and remainder r.

2 Congruence

Let n be a positive integer. Two integers a and b are congruent modulo n if a− b is a multiple of n; that is, if a and b have the same remainder on division by n. In symbols, we write

a ≡ b (mod n). Such a statement is called a congruence, and n is called the modulus of the congruence.

Theorem 3.2 Properties of congruences Let n and k be positive integers, and let a, b, c and d be integers. Then

(a) a ≡ a (mod n); (b) if a ≡ b (mod n), then b ≡ a (mod n); (c) if a ≡ b (mod n) and b ≡ c (mod n), then

a ≡ c (mod n); (d) if a ≡ b (mod n) and c ≡ d (mod n), then

a + c ≡ b + d (mod n); (e) if a ≡ b (mod n) and c ≡ d (mod n), then

ac ≡ bd (mod n); (f) if a ≡ b (mod n), then ak ≡ bk (mod n).

3 For any integer n ≥ 2, Zn = 0, 1, . . . , n− 1.

For a and b in Zn, the operations +n and ×n are defined by:

a +n b is the remainder of a + b on division by n; a×n b is the remainder of a× b on division by n.

The integer n is called the modulus for this arithmetic.

I3

4 Arithmetic in Zn

Addition A1 If a, b ∈ Zn, then

a +n b ∈ Zn. closure A2 If a ∈ Zn, then

a +n 0 = 0 +n a = a. identity A3 If a ∈ Zn, then there is a

number b ∈ Zn such that a +n b = b +n a = 0. inverses

A4 If a, b, c ∈ Zn, then (a +n b) +n c = a +n (b +n c). associativity

A5 If a, b ∈ Zn, then a +n b = b +n a. commutativity

Multiplication M1 If a, b ∈ Zn, then

a×n b ∈ Zn. closure M2 If a ∈ Zn, then

a×n 1 = 1 ×n a = a. identity M4 If a, b, c ∈ Zn, then

(a×n b) ×n c = a×n (b×n c). associativity M5 If a, b ∈ Zn, then

a×n b = b×n a. commutativity

Addition and multiplication D If a, b, c ∈ Zn, then

a×n (b +n c) = a×n b +n a×n c. distributivity

If a, b ∈ Zn and a +n b = 0, then we say that b is the additive inverse of a in Zn, sometimes denoted by −na. If a, b ∈ Zn and a×n b = b×n a = 1, then we say that b is the multiplicative inverse of a in Zn,

−1denoted by a . For some n, the set Zn contains non-zero elements that do not have multiplicative inverses, so in general the inverses property M3 does not hold for multiplication in Zn.

5 Two positive integers a and b have a common factor c, where c is a positive integer, if a and b are both divisible by c. The largest common factor of a and b is usually called their greatest common factor. Two positive integers a and b are said to be coprime, or relatively prime, if their only common factor is 1.

6 Multiplicative inverses

Theorem 3.3 Let n and a be positive integers, with a in Zn. Then a has a multiplicative inverse in Zn if and only if a and n are coprime.

21

I3

Corollary to Theorem 3.3 Let p be a prime number. Then every non-zero element in Zp has a multiplicative inverse in Zp.

If p is prime, then for Zp we can add the following property to the list of properties for multiplication in Zn.

M3 If a ∈ Zp, and a = 0, then a has a multiplicative inverse a−1 ∈ Zp such that

a ×p a−1 = a−1 ×p a = 1. inverses

This property does not hold for Zn if n is not prime.

7 The multiplicative inverse of a in Zn, where a and n are coprime, can be found by using Euclid’s Algorithm as follows. 1. Apply the Division Algorithm repeatedly,

starting by dividing n by a, then a by the remainder of the first division, then the remainder of the first division by the remainder of the second division, and so on, until the remainder 1 is reached.

2. Use the final equation from step 1 to express the final remainder 1 in terms of the previous two remainders.

3. In this expression, substitute for the second-last remainder using the second-last equation from step 1, and rearrange to express 1 in terms of the two remainders previous to the second-last one. Then substitute for the third-last remainder using the third-last equation, and so on, until 1 is expressed in terms of a and n.

4. Rearrange the equation obtained in step 3 to express a multiple of a as a multiple of n plus 1, and deduce the inverse of a in Zn.

In the above steps, ‘in terms of’ two numbers means ‘as a multiple of one of the numbers plus a multiple of the other number’.

8 Linear equations in modular arithmetic

The linear equation

a ×n x = c,

where a, c ∈ Zn, can have no solutions, exactly one solution, or more than one solution x in Zn, as follows. If a and n are coprime, then the equation has exactly one solution in Zn, namely x = a−1 ×n c. If a and n have a common factor d ≥ 2 that is not also a factor of c, then the equation has no solutions in Zn. If a and n have greatest common factor d ≥ 2, and d is a factor of c, then the equation has d solutions in Zn, namely

x = b, x = b + n/d, . . . , x = b + (d − 1)n/d,

where b is the smallest solution.

4 Equivalence relations

1 Relations

We say that ∼ is a relation on a set X if wheneverx, y ∈ X, the statement x ∼ y is either true or false.If x ∼ y is true, then x is related to y.If x ∼ y is false, then x is not related to y and wewrite x y.An equivalence relation on a set X is a relation ∼on X which satisfies the following three properties.E1 reflexive For all x ∈ X,

x ∼ x.

E2 symmetric For all x, y ∈ X, if x ∼ y, then y ∼ x.

E3 transitive For all x, y, z ∈ X, if x ∼ y and y ∼ z, then x ∼ z.

A collection of non-empty subsets of a set is a partition of the set if every two subsets in the collection are disjoint (have no elements in common) and the union of all the subsets in the collection is the whole set. Let ∼ be an equivalence relation defined on a set X; then the equivalence class of x ∈ X, denoted by [[x]], is the set

[[x]] = y ∈ X : x ∼ y. Thus [[x]] is the set of all elements in X related to x.

Theorem 4.1 The equivalence classesassociated with an equivalence relation on aset X have the following properties.(a) Each x ∈ X is in an equivalence class. (b) For all x, y ∈ X, the equivalence classes [[x]]

and [[y]] are either equal or disjoint. Thus the equivalence classes form a partitionof X.

It is sometimes useful to choose a particular element x in each equivalence class and denote the class by [[x]]. The element x that we choose is called a representative of the class.

22

Group Theory Block A

GTA1 Symmetry

1 Symmetry in R2

1 A plane figure is any subset of the plane R2 . A bounded figure in R2 is a figure that can besurrounded by a circle (of finite radius).

An isometry of the plane is a function f : R2 −→ R2

that preserves distances; that is, for all x, y ∈ R2, the distance between f(x) and f(y) is the same as the distance between x and y.A symmetry of a figure F is an isometry mappingF to itself—that is, an isometry f : R2 −→ R2 suchthat f(F ) = F .A symmetry of a plane figure is a one-one and ontofunction.The set of all symmetries of a plane figure F isdenoted by S(F ).

2 Each symmetry of a bounded plane figure is ofone of the following types.• The identity : equivalent to doing nothing to a

figure. • A rotation: specified by a centre and an angle of

rotation. • A reflection : specified by a line—the axis of

symmetry.

The identity is sometimes called the trivial symmetry. It can be regarded as a zero rotation (or zero translation). We always measure angles anticlockwise, and interpret negative angles as clockwise. Although a translation is an isometry, a non-trivial translation cannot be a symmetry of a bounded figure because it does not map the figure onto itself. However, a translation can be a symmetry of an unbounded figure. Two symmetries f and g of a figure F are equal if they have the same effect on F ; that is, f(x) = g(x) for all x ∈ F .

3 Composition of symmetries

Order of composition is important. If f, g ∈ S(F ), then g f may not be equal to f g. That is, in general, composition of symmetries is not commutative.

GTA1

Composition of rotations and reflections follows a standard pattern, as follows.

rotation reflection

rotation rotation reflection reflection reflection rotation

Composing a reflection with itself gives the identity.

4 Properties of symmetries

Let F be a plane figure.The set of symmetries S(F ) of F is closed undercomposition of functions; that is, for all f, g ∈ S(F ),g f ∈ S(F ).The set S(F ) contains an identity symmetry esuch that, for each symmetry f ∈ S(F ),

f e = f = e f.

For each symmetry f ∈ S(F ), there is an inverse symmetry f−1 ∈ S(F ) such that

f f−1 = e = f−1 f.

Composition of symmetries is associative; that is, for all f, g, h ∈ S(F ),

h (g f) = (h g) f.

5 If f f = e, then f−1 = f , and we say that f is self-inverse. All reflections are self-inverse.

6 A regular n-gon is a polygon with n equal edges and n equal angles. In general, a regular n-gon has 2n symmetries: n rotations (through multiples of 2π/n) and n reflections in the axes of symmetry through the centre.

7 Symmetries of the disc

Rotation about the centre through any angle is a symmetry of the disc. Likewise, reflection in any axis through the centre is a symmetry of the disc. Thus the disc has infinitely many rotational and infinitely many reflectional symmetries. We denote a rotation about the centre through an angle θ by rθ, and a reflection in the axis of symmetry making an angle φ with the horizontal axis by qφ, as shown below.

23

GTA1

We restrict the angles for rotations to the interval [0, 2π), and the angles for axes of symmetry to the interval [0, π). So our symmetries are:

rθ: rotation through an angle θ about the centre, for θ ∈ [0, 2π);

qφ: reflection in the line through the centre at an angle φ to the horizontal (measured anticlockwise), for φ ∈ [0, π).

We denote the set of all symmetries of the disc by S(©), read as ‘S-disc’:

S(©) = rθ : θ ∈ [0, 2π) ∪ qφ : φ ∈ [0, π). General formulas for the composition of elements in S(©) are given in the following table.

rθ qθ

The two-line symbol representing f is( ) 1 2 3 · · · n

f(1) f(2) f(3) · · · f(n) .

The order of the columns in the symbol is not important, although we usually use the natural order to aid recognition. To determine g f , the composite of two symmetries g and f written as two-line symbols, we reorder the columns of the symbol for g to make its top line match the order of the bottom line of the symbol for f . We then read off the two-line symbol for the composite g f as the top line of the symbol for f and the bottom line of the symbol for g. To find the inverse of f , we interchange the rows of its two-line symbol. Reordering the columns in the symbol into the natural order is optional but may rφ r(φ+θ) (mod 2π) q( 1

2φ+θ) (mod π) make the inverse easier to recognise. qφ q(φ− 1

2 θ) (mod π) r2(φ−θ) (mod 2π)

2 To form the Cayley table for the elements of a set S(F ) of symmetries, we list the elements of S(F ) across the top and down the left-hand side of a square array, using the same ordering across the top and down the side. Normally we put the identity symmetry e first, as shown below. For any two elements x and y of S(F ), the composite x y is recorded in the cell in the row labelled x and the column labelled y.

e . . . x . . . y . . .

e · · · x · · · y · · ·

. . . · · · x y · · ·

. . .

Note that x is on the left both in the composite andin the border of the table.The leading diagonal of a Cayley table is thediagonal from top left to bottom right.

3 Group axioms

1 A binary operation is a means of combining two elements. Let G be a set and let be a binary operation defined on G. Then (G, ) is a group if the following four axioms G1–G4 hold. G1 closure For all g1, g2 ∈ G,

g1 g2 ∈ G. G2 identity There exists an identity

element e ∈ G such that, for all g ∈ G,

g e = g = e g.

8 Direct and indirect symmetries The symmetries of a plane figure F that we can physically demonstrate without lifting a model out of the plane to turn it over are called direct symmetries. We denote the set of direct symmetries of a figure F by S+(F ). The remaining symmetries are called indirect symmetries: they are the symmetries that cannot be demonstrated physically without lifting a model out of the plane, turning it over and then replacing it in the plane. Rotations and translations are direct symmetries, whereas reflections are indirect symmetries. Composition of direct and indirect symmetries follows a standard pattern, as follows.

direct indirect

direct direct indirect indirect indirect direct

Further properties of direct and indirect symmetries are given in item 5 on page 26. A glide-reflection is a type of indirect symmetry of an unbounded plane figure. It is a composite of a reflection and a translation. Rotations, reflections, translations and glide-reflections are the only possible symmetries of plane figures.

2 Representing symmetries

1 Let f be a symmetry of a polygonal figure F which moves the vertices of the figure F originally at the locations labelled 1, 2, 3, . . . , n to the locations labelled f(1), f(2), f(3), . . . , f(n), respectively.

24

G3 inverses For each g ∈ G, there exists an inverse element g−1 ∈ G such that

g g−1 = e = g−1 g. G4 associativity For all g1, g2, g3 ∈ G,

g1 (g2 g3) = (g1 g2) g3.

A group (G, ) that has the additional property that for all g1, g2 ∈ G,

g1 g2 = g2 g1,

is an Abelian group (or commutative group). A group that is not Abelian is called non-Abelian.

Strategy 3.1 To determine whether (G, ) is a group. guess behaviour, . . . check definition. To show that (G, ) is a group, show that each of the axioms G1, G2, G3 and G4 holds.

To show that (G, ) is not a group, show that any one of the axioms G1, G2, G3 or G4 fails; that is,

show that is not closed on G, or show that there is no identity element in G, or find one element in G with no inverse in G, or show that is not associative.

2 A group (G, ) is a finite group if G is a finite set; otherwise, G is an infinite group.If G is a finite set with exactly n (distinct) elements,then the group (G, ) has order n and we denotethis by writing

| G| = n; otherwise, (G, ) has infinite order.

Examples of finite groups ∗(Z4, +4), (Z5, +5), (Z5, × 5) where Z ∗ = 1, 2, 3, 4 .5

Examples of infinite groups ∗(Z, +), (R ∗ , × ) where R = R − 0 .

3 Uniqueness properties In any group: the identity element is unique; each element has a unique inverse.

4 We can form a Cayley table for any small set G and binary operation defined on G in the same way as for sets of symmetries. (See page 24.) A group table is a Cayley table that represents a group. A Cayley table for (G, ) is a group table if it has all of the following properties. (G1) The table contains only the elements of the

set G; that is, no new elements appear in the body of the table.

GTA1

(G2) A row and a column with the same label repeat the borders of the table. The corresponding element is an identity element, e say.

(G3) The identity element e occurs exactly once in each row and column, and e also occurs symmetrically about the leading diagonal. (This ensures that each element of G has an inverse in G.) (To find the inverse of the element x, look along the row labelled x until you meet the identity e; then x−1 is the label of this column.)

(G4) The operation is associative. (This property is not easy to check from a Cayley table.)

If a group table has the following property, then the group is Abelian:

the table is symmetrical about the leading diagonal.

5 Standard identities For composition of functions, the identity is x −→ x. For addition of real and complex numbers, theidentity is 0.For multiplication of real and complex numbers, theidentity is 1.

6 Standard inverses For composition of functions, the inverse of the function f is the inverse function f −1 . For addition of real and complex numbers, the inverse of x is − x. For multiplication of real and complex numbers, the inverse of x is x−1 = 1/x, provided that x = 0.

7 Standard associative operations The following operations are associative and may be quoted as such:

composition of functions;addition of real and complex numbers,modular addition;multiplication of real and complex numbers, modular multiplication.

(See also page 75, Section 5, item 3.)

8 For four elements a, b, c, d and an associative operation , an expression such as a b c d is unambiguous, and this is true for composites of any finite number of elements. We need not consider the order in which the compositions are carried out, but the order in which the elements appear must be maintained.

25

GTA1

4 Proofs in group theory

1 Properties of inverses In any group (G, ), if g ∈ G and g has inverse g −1 ∈ G,

−1then g has inverse g.

In symbols, we write −1)−1(g = g.

In any group (G, ), with x, y ∈ G, −1 −1(x y)−1 = y x .

2 Properties of group tables

In a group table: • the identity e occurs exactly once in each row

and each column of the table, and in symmetrical positions with respect to the leading diagonal;

• each element of the group occurs exactly once in each row and exactly once in each column.

Warning A Cayley table with the above properties is not necessarily a group table (because the associativity axiom may fail).

3 A group element x is self-inverse if and only if the element on the leading diagonal in the row labelled x in the group table is the identity e.

4 Cancellation laws In any group (G, ) with elements a, b and x:

if x a = x b, then a = b; if a x = b x, then a = b.

5 Symmetry in R3

1 A figure in R3 is any subset of R3 . A bounded figure in R3 is a figure that can be surrounded by a sphere (of finite radius). A bounded non-planar figure with polygonal faces is called a polyhedron. A convex polyhedron is a polyhedron without dents,dimples or spikes.A regular polyhedron (Platonic solid) is a convex polyhedron in which all the faces are congruentregular polygons and each vertex is the junction ofthe same numbers of edges and faces, arranged in thesame way.There are precisely five regular polyhedra:the tetrahedron, which has four triangular faces;the cube, which has six square faces;the octahedron, which has eight triangular faces;the dodecahedron, which has twelve pentagonalfaces;the icosahedron, which has twenty triangular faces.

2 An isometry of R3 is a distance-preserving map f : R3 −→ R3 . A symmetry of a figure F in R3 is an isometry mapping F onto itself—that is, an isometry f : R3 −→ R3 such that f (F ) = F . Two symmetries of a figure F are equal if they have the same effect on F .

3 A rotation of of a figure F in R3 is a symmetry specified by an axis of symmetry (also called axis of rotation), a direction of rotation and the angle through which the figure is rotated. A reflection of F is a symmetry specified by the plane in which the reflection takes place.

4 Direct and indirect symmetries Symmetries of a figure in R3 that we can demonstrate physically with a model (for polyhedra, this means rotations) are called direct symmetries, whereas those that we cannot show physically with the model are called indirect symmetries.

5 Properties of direct and indirect symmetries

The following results apply to figures in R3, and also to figures in R2 . 1. Composition of direct and indirect symmetries

follows a standard pattern, as follows. direct indirect

direct direct indirectindirect indirect direct

2. The set S(F ) of all symmetries of a figure F forms a group under composition.

3. The set S+(F ) of all direct symmetries of a figure F forms a group under composition.

4. If the group S(F ) is finite, then it comprises either • all direct symmetries, or • half direct symmetries and half indirect

symmetries.

5. If the group S(F ) contains n direct and n indirect symmetries, then the n indirect symmetries may be obtained by composing each of the n direct symmetries with any one fixed indirect symmetry.

26

( ) ( )

GTA2

GTA2 Groups and subgroups

1 Groups and subgroups

1 A subgroup of a group (G, ) is a group (H, ),where H is a subset of G.A subgroup has the same binary operation as theparent group, and the same identity element.

6 Counting symmetries of polyhedra

The order of the symmetry group of a polyhedron can be calculated by using one of the following strategies.

Strategy 5.1 To determine the number of symmetries of a regular polyhedron. 1. Count the number of faces. 2. Count the number of symmetries of a face. Then ⎞⎛

number ofsymmetries of ⎠⎝

Theorem 1.1 Let (G, ) be a group withidentity e and let H be a subset of G. Then

regular polyhedron

number of number of= faces × symmetries of face .

Strategy 5.2 To determine the number of

(H, ) is a subgroup of (G, ) if and only if the following three properties hold. SG1 closure For all h1, h2 ∈ H, the

composite h1 h2 ∈ H. SG2 identity The identity element e ∈ H.

symmetries of a non-regular polyhedron. 1. Select one type of face and count the

number of similar faces which are similarly placed in the polyhedron.

2. Count the symmetries of the face within the polyhedron (that is, the symmetries of the face that are also symmetries of the polyhedron).

Then

SG3 inverses For each h ∈ H, the inverse element h−1 ∈ H.

Every group with more than one element has at least two subgroups: the group (G, ) itself, and the trivial subgroup (e, ) consisting of the identity element alone. A subgroup other than the whole group (G, ) is called a proper subgroup. ⎞⎛

number of Strategy 1.2 To determine whether (H, ) is a subgroup of (G, ), where H ⊆ G.

⎝

⎛

⎠symmetries ofpolyhedron ⎞ guess behaviour, . . . check definition.number of ⎞⎛

number of symmetries of face that are also symmetries of

To show that (H, ) is a subgroup, show that⎜⎜⎜⎝

⎟⎟⎟⎠ ⎝ faces of ⎠ × each of the properties SG1, SG2 and SG3 = .

selected type holds. polyhedron To show that (H, ) is not a subgroup, show

that any one of the properties SG1, SG2 or SG3 fails; that is,

show that is not closed on H, or show that e /∈ H,

/or find one element h ∈ H for which h−1 ∈ H.

If H is not a subset of G, then (H, ) cannot be a subgroup of (G, ). 2 The symmetry group S(F ) of a figure F has a subgroup S+(F ), the subgroup of direct symmetries of F . (For a figure F with no indirect symmetries, S+(F ) = S(F ).)

27

GTA2

Strategy 1.3 To find a subgroup of a given symmetry group of a figure, carry out one of the following. • Modify the figure to restrict its symmetry,

for example, by introducing a pattern of lines or shapes, and then determine which of the symmetries of the original figure are symmetries of the new figure.

• Find the symmetries of the figure that fix a particular vertex (or particular vertices).

2 Cyclic groups

1 Let x be an element of a group (G, ). Powers of x are defined inductively, as follows:

0 x = e, the identity element; n = xn−1x ( x)n where n ∈ Z+ .

x−n = x−1

The set of all powers of x is written as 〈x〉 = x k : k ∈ Z;

this is the subset of G generated by x.

2 The previous statements may be ‘translated’ into additive notation.

Multiplicative notation Additive notation

identity e or 1 identity 0x× x = x2 x + x = 2xinverse x−1 inverse −x

nx nx −nx

xs × xt = xs+t

〈x〉 = xk : k ∈ Z(powers)

−nx sx + tx = (s + t)x 〈x〉 = kx : k ∈ Z

(multiples)

3 Generated sets have the following properties: 〈e〉 = e; 〈x −1〉 = 〈x〉;

2if x is self-inverse (i.e. x = e), then 〈x〉 = e, x. 4 Let x be an element of a group (G, ). If n is the least positive integer such that xn = e, then x has order n. Thus the term ‘order’ has two (related) meanings in group theory: the order of an element is defined here, and the order of a group, defined earlier, is the number of its elements.

28

Theorem 2.1 Let x be an element of a group G. If x has order n, then 〈x〉 has exactly n distinct elements. In multiplicative notation,

n−1〈x〉 = e, x, x 2 , . . . , x .In additive notation,〈x〉 = 0, x, 2x, . . . , (n− 1)x.

Theorem 2.2 Let x be an element of agroup (G, ), which may be a finite or aninfinite group. Then

(〈x〉, ) is a subgroup of (G, ).

Usually, we omit the symbol ‘’ and write just 〈x〉 is a subgroup of G.

5 Let x be an element of a group (G, ); then

〈x〉 is a cyclic subgroup of G; 〈x〉 is generated by x; x is a generator of 〈x〉.

If x has order n, then x has finite order, and 〈x〉 is a finite cyclic subgroup of order n.

6 A group (G, ) is a cyclic group if there exists an element x ∈ G such that 〈x〉 = G; if there is no such x, then G is non-cyclic.

Theorem 2.3 Let G be a finite group oforder n. Then G is cyclic if and only if Gcontains an element of order n.

All cyclic groups are Abelian (but not all Abeliangroups are cyclic).

7 Let x be an element of a group (G, ). If there isno positive integer n such that xn = e, then x hasinfinite order, and all the powers of x are distinct.In this case, 〈x〉 is the infinite cyclic subgroupgenerated by x.In multiplicative notation,

−2 −1 0 2 k〈x〉 = . . . , x −k , . . . , x , x , x , x, x , . . . , x , . . .(where x0 = e). In additive notation, 〈x〉 = . . . ,−kx, . . . ,−2x,−x, 0, x, 2x, . . . , kx, . . .

8 The infinite group (S(©), ) has both finite and infinite cyclic subgroups. For example, 〈qφ〉 = e, qφ has order 2, for any φ ∈ [0, π); 〈r2π/n〉 = e, r2π/n, r4π/n, . . . , r2(n−1)π/n has order n,

for any n ∈ N; 〈r1〉 has infinite order.

3 Isomorphisms

1 For any positive integer n, the group table for (Zn, +n) exhibits a pattern of diagonal stripes when we list the elements in the border in the natural order.

2 Two groups (G, ) and (H, ∗ ) are isomorphic if there exists a mapping φ : G −→ H such that both the following statements hold. (a) φ is one-one and onto. (b) For all g1, g2 ∈ G,

φ(g1 g2) = φ(g1) ∗ φ(g2).

Such a function φ is called an isomorphism. We write (G, ) ∼= (H, ∗ ) to denote that the groups (G, ) and (H, ∗ ) are isomorphic. (We sometimes abbreviate this to G ∼= H, with the group operations being understood.)There may be more than one isomorphism mappingG onto H.

3 An isomorphism is one-one and onto, so isomorphic groups (G, ) and (H, ∗ ) have the same order: either (G, ) and (H, ∗ ) are both infinite groups, or (G, ) and (H, ∗ ) are both finite groups

and | G| = | H| . Two finite groups are isomorphic if and only if theirgroup tables can be arranged to exhibit the same pattern.

4 Isomorphic groups have the same structure; for example, any group isomorphic to an Abelian group is Abelian, and any group isomorphic to a cyclic group is cyclic.

GTA2

5 The set of all groups can be divided into classes, called isomorphism classes, as follows. Two groups belong to the same isomorphism class if they are isomorphic, but to different classes otherwise; each group belongs to exactly one class. There are exactly two isomorphism classes for groups of order 4: one contains cyclic groups; the other contains groups in which each element is self-inverse. C4 denotes a typical cyclic group of order 4; K4 denotes a typical group of order 4 in which

each element is self-inverse (referred to as the Klein group).

Strategy 3.1 To show that two groups (G, ) and (H, ∗ ) are isomorphic, show that there is a mapping φ : G −→ H such that: 1. φ is one-one and onto; 2. for all g1, g2 ∈ G,

φ(g1 g2) = φ(g1) ∗ φ(g2). If (G, ) and (H, ∗ ) are finite groups of (the same) small order, it is sufficient to construct the Cayley tables for the two groups and to rearrange one of them to exhibit the same pattern as the other. Then write down a one-one onto mapping φ : G −→ H which matches up the Cayley tables. If (G, ) and (H, ∗ ) are infinite groups or finite groups of (the same) large order, find a suitable mapping φ and show that it has properties 1 and 2.

Strategy 3.2 To show that two finite groups (G, ) and (H, ∗ ) are not isomorphic, try any of the following methods. • Compare the orders | G| and | H| : if

| G| = | H| , then (G, ) (H, ∗ ). • Ascertain whether G and H are cyclic or

Abelian: if one group is Abelian and the other is not, or if one group is cyclic and the other is not, then (G, ) (H, ∗ ).

• If the order is small, compare the entries in the leading diagonals of the group tables for G and H. For example, count the number of times the identity element appears and count the number of different elements that appear. If either of these counts differs between the two groups, then (G, ) (H, ∗ ).

29

( )

( )

GTA2

6 Isomorphisms of cyclic groups

Theorem 3.1 Two cyclic groups of the same order are isomorphic.

Strategy 3.3 To find an isomorphism between two finite cyclic groups G and H of the same order. 1. Find a generator g of G and a generator h

of H. 2. Construct the following isomorphism φ:

φ : G −→ H

g −→ h k g −→ hk , for k = 2, 3, . . . .

7 The symbol Cn denotes a typical cyclic group of order n, generated by x:

2 nCn = 〈 x〉 = e, x, x , . . . , x n−1 : x = e .

8 Properties of isomorphisms

The relation is isomorphic to, denoted by ∼=, is an equivalence relation on the set of all groups. If (G, ) and (H, ∗ ) are groups with identities eG

and eH , respectively, and φ : (G, ) −→ (H, ∗ ) is an isomorphism, then φ has the following properties. 1. identity φ matches the identity elements:

φ(eG) = eH . 2. inverses φ matches inverses:

for each g ∈ G, φ g−1 = (φ(g))−1 .

3. powers φ matches ‘powers’: for each g ∈ G and each k ∈ Z,

φ gk = (φ(g))k .

4 Groups from modular arithmetics

1 Additive modular arithmetics

Theorem 4.1 For each n ∈ N, (Zn, +n) is a cyclic group of order n and 1 is a generator.

Theorem 4.2 Let r be a non-zero element of Zn. Then r is a generator of (Zn, +n) if and only if r is coprime to n.

Theorem 4.3 The group (Zn, +n) has exactly one cyclic subgroup of order m for each divisor m ∈ N of n. This subgroup is either 〈 0〉 (in the case where m = 1), or is generated by q, where mq = n.

Theorem 4.4 Let (G, ) be a cyclic group (finite or infinite). Then all the subgroups of (G, ) are cyclic.

Theorems 4.3 and 4.4 together give us a way of finding all the subgroups of a given finite cyclic group: we just find all its distinct cyclic subgroups.

2 Multiplicative modular arithmetics

We define ∗ Z = Zn − 0 = 1, 2, 3, . . . , n− 1 .n

Theorem 4.5 Let p be a prime number. ∗Then (Zp,× p) is a group of order p− 1.

Theorem 4.6 For each n ∈ N, the set of all numbers in Zn that are coprime to n forms a group under × n.

30

( )

GTA3 Permutations

1 Permutations

1 A permutation of a finite set S is a one-one function from S onto S.We refer to the elements of S as the ‘symbols beingpermuted’. Usually we take S = 1, 2, . . . , n.2 A permutation can be written as a two-linesymbol. An example of a permutation of1, 2, 3, 4, 5, 6, 7, 8 written in this way is

1 2 3 4 5 6 7 8 f = 4 6 8 3 1 2 7 5

.

Here 1 maps to 4, 2 maps to 6, and so on.

The same permutation f can also be written in cycle form:

f = (1 4 3 8 5)(2 6)(7). This notation indicates that f maps the symbols as follows.

We say that f is the product of the disjoint cycles (1 4 3 8 5), (2 6) and (7). Here disjoint means that each symbol belongs to only one cycle. As a cycle has no particular starting point, we can write any of its symbols down first. Also, disjoint cycles can be written in any order. Thus, for example,

f = (6 2)(7)(3 8 5 1 4) or f = (7)(5 1 4 3 8)(2 6). However, we usually write the smallest symbol in each cycle first, and arrange the cycles with their smallest symbols in increasing order.

Strategy 1.1 To find the cycle form of a permutation f . 1. Choose any symbol (such as 1) and find its

successive images under f until you encounter the starting symbol again.

2. Write these symbols as a cycle. 3. Repeat the process starting with any

symbol which has not yet been used, until there are no symbols left.

GTA3

Theorem 1.1 Any permutation can be broken down into disjoint cycles. This cycle form is unique, apart from the choice of starting symbol in each cycle and the order in which the cycles are written.

3 When a cycle consists of a single symbol, the permutation maps that symbol to itself. We say that the symbol is fixed by the permutation. For example,

f = (1 6 3)(2)(4 7)(5) fixes both 2 and 5. A permutation is written in cycle form when it iswritten as a product of disjoint cycles.When there is no ambiguity concerning which set ofsymbols is being permuted, it is customary to omitfixed symbols from the cycle form. For example, wewrite the above permutation f as

f = (1 6 3)(4 7). When working with permutations in cycle form, it is customary to denote the identity permutation, which fixes every symbol, by e.

4 The order in which permutations are composed is important: if f and g are permutations of 1, 2, . . . , n, then in general f g and g f are different permutations.

Strategy 1.2 To find the composite g f of two permutations given in cycle form. 1. Consider the smallest symbol, 1 say. Find

f(1) and g(f(1)), and write the result, x say, next to 1 in a cycle:

(1 x . . .), where x = g(f(1)). 2. Starting with the symbol x, repeat the

process to obtain the next symbol g(f(x)) in the cycle.

3. Continue repeating the process until the next symbol found is the original symbol 1. This completes the cycle.

4. Starting with the smallest symbol not yet placed in a cycle, repeat steps 1 to 3 until no more unplaced symbols remain.

Any permutation is equal to the composite of its disjoint cycles; for example, if f = (1 3)(2 5 6)(7 8), then f = (1 3) (2 5 6) (7 8).

31

GTA3

5 The inverse f−1 of a permutation f undoes what f does; that is, if f maps x to y, then f−1

maps y to x.

Strategy 1.3 To find the inverse of a permutation which is given in cycle form, reverse the order of the symbols in each cycle.

For example, if f = (1 3 5 6)(2 4), then

f−1 = (6 5 3 1)(4 2) = (1 6 5 3)(2 4).

6 The symmetric group Sn

Theorem 1.2 The set Sn of all permutations of the set 1, 2, 3, . . . , n is a group under composition.

The group Sn is called the symmetric group of degree n.

Theorem 1.3 The symmetric group Sn has order n!.

7 A permutation group is a subgroup of Sn, for some positive integer n. The number of symbols involved, n, is called the degree of the permutation group.

8 A permutation whose cycle form consists of a single cycle permuting r symbols (with all other symbols fixed) is called an r-cycle, or a cycle of length r. A 2-cycle is also called a transposition. For example, in S6,

(1 3 5 6) is a 4-cycle, (4 6) is a transposition.

Theorem 1.4 An r-cycle has order r.

The least common multiple of a set of positive integers is the smallest positive integer divisible by each number in the set. For example, 12 is the least common multiple of 2, 3, 4.

Theorem 1.5 The order of a permutation is the least common multiple of the lengths of its cycles.

9 Two permutations in Sn have the same cycle structure if, for each r ≥ 1, they have the same number of disjoint r-cycles. There are three different cycle structures in S3.

Cycle structure Elements of S3 Description

(–)(–)(–) (1)(2)(3) = e identity (– –)(–) (1 2), (1 3), (2 3) transpositions (– – –) (1 2 3), (1 3 2) 3-cycles

There are five different cycle structures in S4.

Cycle Elements of S4 Description structure

(–)(–)(–)(–) (1)(2)(3)(4) = e identity (– –)(–)(–) (1 2), (1 3), (1 4), transpositions

(2 3), (2 4), (3 4) (– – –)(–) (1 2 3), (1 2 4), 3-cycles

(1 3 4), (1 3 2), (1 4 2), (1 4 3), (2 3 4), (2 4 3)

(– – – –) (1 2 3 4), (1 2 4 3), 4-cycles (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2)

(– –)(– –) (1 2)(3 4), (1 3)(2 4), products of (1 4)(2 3) 2-cycles

2 Even and odd permutations

1 When we write a permutation in cycle form, the order in which the cycles are written does not matter, because they are disjoint. However, we cannot reorder cycles in composites in general, because composition of permutations is not a commutative operation.

2 Any permutation may be expressed as a composite of transpositions; such an expression is not unique.

Strategy 2.1 To express a cycle(a1 a2 a3 . . . ar ) as a composite oftranspositions, write the transpositions

(a1 a2), (a1 a3), (a1 a4), . . . , (a1 ar ) in reverse order and form their composite. Thus

(a1 a2 a3 . . . ar )= (a1 ar ) (a1 ar−1) · · · (a1 a3) (a1 a2).

To express a permutation as a composite of transpositions, write it in cycle form, use the fact that it is equal to the composite of its disjoint cycles, and apply Strategy 2.1 to each of the cycles.

32

Theorem 2.1 Parity Theorem A permutation cannot be written both as a composite of an even number of transpositions and as a composite of an odd number of transpositions.

3 A permutation is even if it can be expressed as a composite of an even number of transpositions; a permutation is odd if it can be expressed as a composite of an odd number of transpositions. The evenness/oddness of a permutation is called its parity.

Theorem 2.2 In the group Sn, an r-cycle is an even permutation, if r is odd,an odd permutation, if r is even.

In particular, a transposition is an oddpermutation and the identity element is an even permutation.

Strategy 2.2 To determine the parity of a permutation. 1. Express the permutation as a composite of

cycles (either disjoint or not). 2. Find the parity of each cycle, using the rule:

even, if r is odd,an r-cycle is odd, if r is even.

3. Combine the even and odd parities usingthe following table.

+ even odd

even even odd odd odd even

4 Alternating group An

Theorem 2.3 The set An of all even permutations of the set 1, 2, 3, . . . , n is a subgroup of the symmetric group Sn.

The group An of all even permutations of 1, 2, . . . , n is called the alternating group of degree n.

Theorem 2.4 For n ≥ 2, the alternating group An has order 1

2 (n!).

GTA3

The group A4 comprises the twelve elements e, (1 2 3), (1 3 2), (1 4)(2 3),

(2 4 3), (1 3 4), (1 4 2), (1 3)(2 4), (2 3 4), (1 4 3), (1 2 4), (1 2)(3 4).

These correspond to the twelve rotations of the regular tetrahedron.

3 Conjugacy in Sn

1 The elements x, y ∈ Sn are conjugate in Sn if there exists an element g ∈ Sn such that

−1 y = g x g .

The element g is a conjugating element that conjugates x to y, and y is the conjugate of x by g.

2 If x and g are permutations in Sn, then the conjugate g x g−1 of x by g can be obtained by ‘renaming’ each symbol in x using g; that is, we replace each symbol in the cycle form of x by its image under g. For example, for x = (1 4 3)(2 6) and g = (1 3 2 4 5), the renaming is as follows:

(1 4 3) (2 6) g ↓ ↓ ↓ ↓ ↓

(g(1) g(4) g(3)) (g(2) g(6)) = (3 5 2) (4 6)

so

g x g −1 = (3 5 2)(4 6) = (2 3 5)(4 6).

3 If x and y are elements of Sn with the same cycle structure, then there is an element g ∈ Sn which conjugates x to y.

Strategy 3.1 To find g ∈ Sn such that y = g x g−1, where x, y ∈ Sn have the same cycle structure. 1. Match up the cycles of x and y so that

cycles of equal lengths correspond.x = (∗∗ . . . ∗) (∗∗ . . . ∗) . . . (∗) (∗)

g ↓ ↓ ↓ ↓ ↓ y = (∗∗ . . . ∗) (∗∗ . . . ∗) . . . (∗) (∗)

2. Read off the ‘renaming permutation’ g from the above ‘two-line symbol’.

For example, with x = (1 2 4)(3 5) and y = (1 4)(2 5 3) in S5, we obtain

(1 2 4) (3 5) g ↓ ↓ ↓ ↓ ↓

(2 5 3) (1 4) which gives g = (1 2 5 4 3).

In general, there are many possible matchings and therefore many possibilities for g.

33

( )

GTA3

Theorem 3.1 Two elements of Sn are conjugate in Sn if and only if they have the same cycle structure.

4 Let H be a subgroup of Sn and let g ∈ Sn. Then −1 isthe set of elements of G of the form g h g

denoted by gHg−1. That is,

gHg −1 = g h g −1 : h ∈ H .

Theorem 3.2Conjugate Subgroups TheoremLet H be a subgroup of Sn and let g ∈ Sn; then gHg−1 is also a subgroup of Sn. The groups H and gHg−1 are conjugate subgroups in Sn.

Strategy 3.2 To find the subgroup gHg−1 , given H and g, calculate g h g−1, for each h ∈ H, by using g to ‘rename’ the symbols in h.

4 Subgroups of S4

1 The following table gives the number of subgroups of S4 of each order.

Order Number of Description subgroups

1 2

1 9

eall cyclic

3 4 all cyclic 4 7 3 cyclic; 4 Klein 6 4 all isomorphic to S( ) 8 3 all isomorphic to S()

12 1 A4

24 1 S4

If the vertices of the tetrahedron are labelled 1, 2, 3 and 4, then each element of S4 represents a symmetry of the tetrahedron. Thus S4 and S(tet) are isomorphic groups.

2 An efficient way to find all the cyclic subgroups of any reasonably small group is to first list the elements of the group according to their orders, and then find the distinct cyclic subgroups that they generate.

A useful way to find a non-cyclic subgroup of Sn is to draw a figure, labelled at suitable locations with some or all of the symbols 1, 2, . . . , n, such that the symmetries of the figure can be represented by permutations of the labels. The symmetry group of the figure is then a subgroup of Sn. Once we have found a subgroup of a symmetric group by realising it as the symmetry group of a figure, we can often find another subgroup of the same symmetric group, isomorphic to the first subgroup, by relabelling the figure. (Here we are finding subgroups conjugate to the first subgroup.)

5 Cayley’s Theorem

Theorem 5.1 Cayley’s Theorem Every finite group is isomorphic to a permutation group.

To determine a permutation group isomorphic to a given finite group G = g1, g2, g3, . . . , gn , do the following. (The symbol in composites is omitted here.) 1. Write down the Cayley table of G. 2. With each element x ∈ G, associate the two-line symbol Px obtained from the group table of G by taking the top row of the table as the top line of the symbol and the row of x as the bottom line of the symbol.

g1 g2 g3 . . . gn

g1 . . . . . . x xg1 xg2 xg3 . . . xgn . . . . . .

gn

3. This gives a permutation with the two-line symbol

g1 g2 g3 . . . gn column headings Px =

xg1 xg2 xg3 . . . xgn row of x.

4. The composite of two such permutations is given by

Px Py = Pxy ,

which shows that the constructed permutations combine in the same way as the corresponding original elements: the group table for the constructed group has the same pattern as the original group table, with each element g replaced by Pg .

Thus the mapping g −→ Pg is an isomorphism between the two groups, and the set of permutations Pg : g ∈ G forms a group isomorphic to G.

34

GTA4 Cosets and Lagrange’s Theorem

1 Cosets

1 Let H be a subgroup of a group (G, ) and let g be an element of G. The coset gH of H in G is the set of elements of G of the form g h, where h ∈ H. That is,

gH = g h : h ∈ H, which is the set obtained by composing each elementof H with g on the left.If H is finite, say

H = h1, h2, . . . , hm, then

gH = g h1, g h2, . . . , g hm. 2 Properties of cosets

1. For each element g and each subgroup H of a finite group, the coset gH has the same number of elements as H.

2. For each element g and each subgroup H, the element g lies in the coset gH.

3. One of the cosets gH is H itself. 4. Any two cosets g1H and g2H are either the same

set or are disjoint.

Theorem 1.1 Let H be a subgroup of a group G. Then the cosets of H form a partition of G.

Two cosets g1H and g2H are either the same or have no elements in common, thus:

if g2 ∈ g1H, then g2H = g1H; if g2H = g1H, then g2 ∈ g1H and g1 ∈ g2H.

GTA4

Strategy 1.1 To partition a finite group G into cosets of a given subgroup H. 1. Take H as the first coset. 2. Choose any element g not yet assigned to a

coset and determine the coset gH to which g belongs.

3. Repeat step 2 until every element of G has been assigned to a coset.

3 If (G, +) is an additive group with subgroup H, then we write the coset of H in G containing the element g as g + H rather than gH. Thus

g + H = g + h : h ∈ H. 4 The definition of coset and Theorem 1.1 are applicable to any group, finite or infinite. A partition of an infinite group into cosets may involve infinitely many cosets. In that case, Strategy 1.1 can be used to discover more and more cosets, but not all of them. However, it may be possible to find a general form for them.

2 Lagrange’s Theorem for finite groups

1 Cosets can be used to prove the following result.

Theorem 2.1 Lagrange’s Theorem Let G be a finite group and let H be any subgroup of G. Then the order of H divides the order of G.

Lagrange’s Theorem allows us to write down all the possible orders for subgroups of a finite group G—namely, all the positive divisors of the order of G. Thus, if the number m does not divide the order of G, then G does not have a subgroup of order m. Warning The converse of Lagrange’s Theorem is false. Lagrange’s Theorem does not assert that if m is a positive divisor of the order of a group G, then G has a subgroup of order m.

2 Let H be a subgroup of a group G. Then the number of distinct cosets of H in G is called the index of H in G. A subgroup H of an infinite group may have finite index; otherwise, H is said to have infinite index. If G is a finite group, then the index k of H in G is the order of G divided by the order of H:

k = |G|/|H|.

35

GTA4

3 Consequences of Lagrange’s Theorem

Corollary 1 Let g be an element of a finite group G. Then the order of g divides the order of G.

Corollary 2 If G is a group of prime order, then G is a cyclic group. The only subgroups of G are e and G, and each element of G other than e generates G.

Corollary 3 If G is a group of prime order p, then G is isomorphic to the cyclic group Zp.

3 Groups of small order

1 There is only one isomorphism class for groups of order 1. The only element of such a group is the identity element. For each prime p, there is only one isomorphism class for groups of order p, and all the groups in this class are cyclic.

Theorem 3.1 Let G be a group in which each element except the identity has order 2. Then G is Abelian.

Theorem 3.2 Let G be a group, with order greater than 2, in which each element except the identity has order 2. Then the order of G is a multiple of 4.

Theorem 3.3 Let G be a group of even order. Then G contains an element of order 2.

2 There are two isomorphism classes for groups of order 4:

one contains the cyclic group C4; the other contains the Klein group K4.

All groups of order 4 are Abelian.

Let G be a group of order 4.If G contains an element of order 4 (there would betwo such elements), then G ∼= C4. If all elements are self-inverse, then G ∼= K4.

3 There are two isomorphism classes for groups of order 6:

one contains the cyclic group C6; the other contains the non-Abelian group S().

Let G be a group of order 6. If the group table is symmetrical about the leading diagonal (G is Abelian), then G ∼= C6. If the group table is not symmetrical about the leading diagonal (G is non-Abelian), then G ∼= S().

4 There are five isomorphism classes for groups of order 8.

Abelian groups Class 1 contains the cyclic groups; these have onlyone element of order 2.Class 2 contains groups in which all the non-identityelements have order 2.Class 3 contains groups with only three elements oforder 2.

Non-Abelian groups Class 4 contains groups with five elements of order 2 (e.g. S()). Class 5 contains groups with only one element of order 2.

Strategy 3.1 To determine the isomorphism class of a group G of order 8. 1. Determine whether G is Abelian. 2. Find the number of elements of G which

have order 2.3. Identify the class from the following table.

Is G Abelian? Number of Class elements of order 2

Yes 1 1 Yes 7 2 Yes 3 3 No 5 4 No 1 5

36

4 Normal subgroups

1 Let H be a subgroup of a group (G, ) and let g be an element of G. The right coset Hg is the set of elements of G of the form h g, where h ∈ H. That is,

Hg = h g : h ∈ H. It is the set obtained by composing each elementof H with g on the right.If H is finite, say

H = h1, h2, . . . , hm, then

Hg = h1 g, h2 g, . . . , hm g. Strictly speaking, the sets that we have been calling cosets up to this point should have been called left cosets, to distinguish them from right cosets. In general, left and right cosets are different sets. If G is an additive group with subgroup H, then we denote the right coset of H in G containing the element g by H + g. Thus

H + g = h + g : h ∈ H. 2 Properties of right cosets

1. For each element g and each subgroup H of a finite group, the right coset Hg has the same number of elements as H.

2. For each element g and each subgroup H, the element g lies in the right coset Hg.

3. One of the right cosets Hg is H itself. 4. Any two right cosets Hg1 and Hg2 are either the

same set or are disjoint. That is, the body of the table splits into ‘blocks’, where each block is an |N | by |N | array of entries all from a single coset.

Strategy 4.1 To partition a finite group G into right cosets of a given subgroup H. 1. Take H as the first coset. 2. Choose any element g not yet assigned to a

right coset and determine the right coset Hg to which g belongs.

3. Repeat step 2 until every element of G has been assigned to a right coset.

GTA4

3 Let G be a group and let H be a subgroup of G. Then H is a normal subgroup of G if the left and right partitions of G into cosets of H are the same. That is, H is normal in G if, for each element g ∈ G,

gH = Hg.

The condition gH = Hg means that the sets gH and Hg contain the same elements; it does not mean thatgh = hg for all h ∈ H.We use the phrases ‘H is a normal subgroup of G’and ‘H is normal in G’ interchangeably.

Theorem 4.1 Let G be a group; then (a) the identity subgroup e is a normal

subgroup of G;(b) the whole group G is a normal subgroup

of G.

Theorem 4.2 In an Abelian group, every subgroup is normal.

Theorem 4.3 Let H be a subgroup of index 2 in a group G. Then H is a normal subgroup of G.

Corollary For all n ≥ 2, the alternating group An is a normal subgroup of the symmetric group Sn.

5 Quotient groups

1 Let A and B be subsets of a group (G, ). Then the binary operation ., called set composition, is defined by

A . B = a b : a ∈ A, b ∈ B; that is, A . B is the subset of G obtained by composing each element of A with each element of Bon the right.For an additive group (G, +), we write + ratherthan . for set composition. Thus

A + B = a + b : a ∈ A, b ∈ B.

37

GTA4

Theorem 5.1 Let N be a normal subgroup of a group G. Then, for all a, b ∈ G,

aN . bN = (a b)N.

Theorem 5.2 Let N be a normal subgroup of a group G. Then the set of cosets of N in G, with the operation of set composition, is a group.

This group is called the quotient group of G by N , and is denoted by G/N .The identity element of G/N is the coset eN = N ,and the inverse of gN is g−1N .If G is a finite group, then the order of G/N is thenumber of cosets of N in G; that is,

|G/N | = |G|/|N |. 2 Let N be a normal subgroup of a (finite) group G and let the elements of G be listed in the borders of its group table in the following order: elements of N , then elements of another coset of N , then elements of a third coset of N , and so on. Then the group table of G ‘blocks’. That is, the body of the table splits into ‘blocks’, where each block is an |N | by |N | array of entries all from a single coset. The resulting table of blocks is essentially the group table of G/N .

Strategy 5.1 To find a group isomorphic to a finite quotient group G/N , where N is a normal subgroup of a group (G, ). 1. Calculate |G/N | = |G|/|N | = k, say. 2. Determine the k cosets aN by choosing

different elements a until all the elementsof G are assigned to cosets.

3. Construct the k × k group table of G/N by composing each pair of cosets, using the rule

aN . bN = (a b)N

= coset containing a b. 4. By inspection of the group table, identify a

familiar group (from the course) isomorphic to G/N .

In the group table of G/N , avoid writing a given coset in more than one way. For example, if cN = dN , then denote every occurrence of this coset by cN (or every occurrence by dN ).

6 Quotient groups of infinite groups

1 The subgroups of Z are

0, Z, and

nZ = 〈n〉 = . . . , −3n, −2n, −n, 0, n, 2n, 3n, . . ., for each n ≥ 2.

They are all cyclic.

Theorem 6.1 For each integer n ≥ 2, the quotient group Z/nZ is a cyclic group generated by 1 + nZ and so

Z/nZ ∼= Zn.

2 Let m be a positive integer. We define addition modulo m on the interval [0, m) by

x + y, if x + y < m, x +m y =

x + y − m, if x + y ≥ m.

The interval [0, m) forms a group under +m.

3 The fractional part of a real number x is x − [x], where [x] is the largest integer not exceeding x. The fractional part of any real number lies in [0, 1).

Theorem 6.2 The quotient group R/Z is isomorphic to ([0, 1), +1), the group of real numbers in the interval [0, 1), under addition modulo 1.

The quotient group R/Z has infinite order. We have

R/Z = r + Z : r ∈ [0, 1), with binary operation

(x + Z) + (y + Z) = (x +1 y) + Z.

38

√

√

Linear Algebra Block

LA1 Vectors and conics

1 Coordinate geometry: points, planes and lines

1 The plane, together with an origin and a pair of x-, y-axes, is often called two-dimensional Euclidean space, denoted by R2 .

2 Equation of a line The general equation of a line in R2 is

ax + by = c,

where a, b and c are real, and a and b are not bothzero.Two distinct lines with equations y = m1x + c1 andy = m2x + c2, where m1 and m2 are both non-zero,are:• parallel if and only if m1 = m2 and c1 = c2; • perpendicular if and only if m1m2 = −1.

Two arbitrary lines in R2 may have a single point of intersection, be parallel, or coincide.

3 Distance formula in R2 The distance between two points (x1, y1) and (x2, y2) in R2 is

(x2 − x1)2 + (y2 − y1)2 .

4 Three-dimensional space, together with an origin and a set of x-, y- and z-axes, is often called three-dimensional Euclidean space, denoted by R3 .

5 Equation of a plane The general equation of a plane in R3 is

ax + by + cz = d,

where a, b, c and d are real, and a, b and c are not all zero. Two arbitrary planes in R3 may intersect, be parallel, or coincide.

6 Distance formula in R3 The distance between two points (x1, y1, z1) and (x2, y2, z2) in R3 is

(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 .

2 Vectors

1 Vectors and scalars

A vector is a quantity that is determined by itsmagnitude and direction. A scalar is a quantity thatis determined by its magnitude.The length (magnitude) of a vector v is denotedby ‖v‖.

LA1

The zero vector is the vector whose magnitude is zero, and whose direction is arbitrary. It is denoted by 0. Two vectors a and b are equal if

they have the same magnitude (‖a‖ = ‖b‖) and

they are in the same direction.

We write a = b. The negative of a vector v is the vector with the same magnitude as v, but the opposite direction.It is denoted by −v.

2 Scalar multiple of a vector Let k be a scalarand v a vector. Then kv is the vector whosemagnitude is |k| times the magnitude of v, that is, ‖kv‖ = |k| ‖v‖, and whose direction is

the direction of v if k > 0, the direction of −v if k < 0.

If k = 0, then kv = 0.

3 Addition of vectors

Triangle Law for addition of vectors The sum p + q of two vectors p and q isobtained as follows.1. Starting at any point, draw the vector p. 2. Starting from the finishing point of the

vector p, draw the vector q. Then the sum p + q is the vector from the starting point of p to the finishing point of q.

Parallelogram Law for addition of vectors The sum p + q of two vectors p and q is obtained as follows. 1. Starting at the same point, draw the vectors

p and q. 2. Complete the parallelogram of which these

are adjacent sides. Then the sum p + q is the vector from the starting point of p and q to the opposite corner of the parallelogram.

39

LA1

The difference p − q of two vectors p and q is p − q = p + (−q).

4 A unit vector is a vector of magnitude 1 unit. In R2, the vectors i and j are unit vectors in the positive directions of the x- and y-axes, respectively. Any vector p in R2 can be expressed as a sum of the form

p = a1i + a2j, for some real numbers a1 and a2;

often we write p = (a1, a2). The numbers a1 and a2 are the components of p in the x- and y-directions, respectively. In R3, the vectors i, j and k are unit vectors in the positive directions of the x-, y- and z-axes, respectively. Any vector p in R3 can be expressed as a sum of the form

p = a1i + a2j + a3k, for some real numbers a1, a2 and a3;

often we write p = (a1, a2, a3). The numbers a1, a2 and a3 are the components of p in the x-, y- and z-directions, respectively.

Equality Two vectors, both in R2 or both in R3 ,are equal if and only if their correspondingcomponents are equal.

Zero vectorThe zero vector in R2 is 0 = 0i + 0j = (0, 0).The zero vector in R3 is 0 = 0i + 0j + 0k = (0, 0, 0).

5 Addition of vectors To add vectors in R2 orin R3 given in component form, add theircorresponding components:

(a1, a2) + (b1, b2) = (a1 + b1, a2 + b2); (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3).

Negative of a vector To find the negative of a vector in R2 or in R3 given in component form, take the negatives of its components: −(a1, a2) = (−a1,−a2); −(a1, a2, a3) = (−a1,−a2,−a3).

Subtraction of vectors To subtract a vector in R2 or in R3 given in component form, subtract its corresponding components:

(a1, a2) − (b1, b2) = (a1 − b1, a2 − b2); (a1, a2, a3) − (b1, b2, b3) = (a1 − b1, a2 − b2, a3 − b3).

6 Multiplication by a scalar To multiply a vector given in component form in R2 or in R3 by a real number k, multiply each component in turn by k:

k(a1, a2) = (ka1, ka2);k(a1, a2, a3) = (ka1, ka2, ka3).

7 The vector space R2 is the set of ordered pairs of real numbers with the operations of addition and multiplication by a scalar defined as follows:

(a1, a2) + (b1, b2) = (a1 + b1, a2 + b2); k(a1, a2) = (ka1, ka2), where k ∈ R.

Similarly, the vector space R3 is the set of ordered triples of real numbers with analogous operations of addition and multiplication by a scalar.

8 The position vector p = a1i + a2j (often written as p = (a1, a2), for brevity) is the vector in R2 whose starting point is the origin and whose finishing point is the point with Cartesian coordinates (a1, a2).

A position vector in R3 is defined similarly.

9 Vector form of the equation of a line

The equation of the line through the points with position vectors p and q is

r = λp + (1 − λ)q, where λ ∈ R.

Section Formula The position vector r of the point that divides the line joining the points with position vectors p and q in the ratio (1 − λ) : λ is

r = λp + (1 − λ)q.

The position vector of the midpoint of this line 1segment is r = 2 (p + q).

3 Dot product

1 The dot product of the vectors u and v in R2 or R3 is

u . v = ‖u‖ × ‖v‖ × cos θ, where ‖u‖ and ‖v‖ denote the lengths of the vectors u and v, and θ is the angle between them.

2 The length of the vector v in terms of the dot product is given by √ ‖v‖ = v . v.

The unit vector in the same direction as v is v

v = . ‖v‖

40

( )

3 If two vectors u and v are orthogonal (perpendicular), then

u . v = 0. Conversely, if u . v = 0, then

either u or v is 0, or u is perpendicular to v.

4 Properties of the dot product

Let u, v and w be vectors, and let α be any real number. Then the following properties hold. Symmetry: u . v = v . u

Multiples: (αu) . v = u . (αv) = α(u . v) Distributivity: u . (v + w) = u . v + u . w,

(u + v) . w = u . w + v . w.

5 The angle θ between two vectors u and v is given by

u . v cos θ = . ‖u‖ × ‖v‖

6 The projection of a vector v onto a vector u is u . v ‖v‖ × cos θ = . ‖u‖

7 Dot product of vectors in component form

In R2, let u = (x1, y1) = x1i + y1j and v = (x2, y2) = x2i + y2j; then

u . v = x1x2 + y1y2.

In R3, let u = (x1, y1, z1) = x1i + y1j + z1k and v = (x2, y2, z2) = x2i + y2j + z2k; then

u . v = x1x2 + y1y2 + z1z2.

8 A vector that is perpendicular to all the vectors in a given plane is called a normal vector to the plane. A normal vector n does not determine a plane uniquely, as there are infinitely many planes that have n as a normal; these planes are parallel to one another. However, if we specify both a normal vector and a point that lies in the plane, then the plane is determined uniquely.

9 Vector form of the equation of a plane

Theorem 3.1 The equation of the plane that contains the point (x1, y1, z1) and has n = (a, b, c) as a normal is

ax + by + cz = d,

where d = ax1 + by1 + cz1.

Corollary The equation of the plane that contains the point (x1, y1, z1) and has n = (a, b, c) as a normal is

x . n = p . n,

where x = (x, y, z) and p = (x1, y1, z1).

LA1

4 Conics

1 Non-degenerate conic sections are parabolas,ellipses and hyperbolas; degenerate conic sectionsare a single point, a single line and a pair of lines.The ellipse and hyperbola each have a centre: there is a point about which rotation through π is asymmetry of the conic.The hyperbola has two lines, called asymptotes,that it approaches.(The non-degenerate conics are illustrated onpage 101.)

2 Circles in R2

Theorem 4.1 The equation of a circle in R2

with centre (a, b) and radius r is 2(x− a)2 + (y − b)2 = r .

Theorem 4.2 An equation of the form

x 2 + y 2 + fx + gy + h = 0

represents a circle with

centre −12 g2 f,−1

and √ 1 f2 + 1

4 g2 − hradius 4

if and only if 1 f2 + 1 4 g

2 − h > 0.4

3 The parabola, ellipse and hyperbola can be defined as the set of points P in the plane that satisfy the following condition: the distance of P from a fixed point is a constant multiple e of the distance of P from a fixed line. The fixed point is the focus of the conic, the fixed line is its directrix, and e is its eccentricity. A non-degenerate conic is

an ellipse if 0 ≤ e < 1, a parabola if e = 1, a hyperbola if e > 1.

4 Parabola in standard form

A parabola in standard form has equation

y 2 = 4ax, where a > 0. It can be described by the parametric equations

2 x = at , y = 2at (t ∈ R). It has focus (a, 0) and directrix x = −a; its axis isthe x-axis and its vertex is the origin.A chord that passes through the focus is a focalchord.See the following diagram.

41

LA1

5 Ellipse in standard form

An ellipse in standard form has equation 2 2x

+ y

= 1, a2 b2

where a ≥ b > 0, b2 = a2(1 − e2), 0 ≤ e < 1. It can be described by the parametric equations

x = a cos t, y = b sin t (t ∈ R). If e > 0, it has foci (±ae, 0) and directrices x = ±a/e; its major axis is the line segment joining the points (±a, 0), and its minor axis is the linesegment joining the points (0, ±b).If e = 0, the ellipse is a circle; the single focus is atthe centre of the circle and the directrix is‘at infinity’.

6 Hyperbola in standard form

A hyperbola in standard form has equation 2 2x y− = 1,

a2 b2

where b2 = a 2(e 2 − 1), e > 1. It can be described by the parametric equations

x = a sec t, y = b tan t

√ Rectangular hyperbola (e = 2) A hyperbola whose asymptotes are at right angles iscalled a rectangular hyperbola.If we use the asymptotes as new x- and y-axes(instead of the original x- and y-axes), then theequation of the hyperbola can be written in the formxy = c2, for some positive number c. A rectangularhyperbola has the origin as its centre, and the x- and y-axes as its asymptotes. It can be described by theparametric equations

c x = ct, y = , where t = 0.

t

7 General equation of a conic

Theorem 4.3 Any conic has an equation of the form

Ax2 + Bxy + Cy2 + Fx + Gy + H = 0, (∗) where A, B, C, F , G and H are real numbers, and A, B and C are not all zero. Conversely, the set of all points in R2 whose coordinates (x, y) satisfy an equation of the form (∗) is a conic.

(t ∈ [−π, π] excluding −π 2 and π).2

It has foci (±ae, 0) and directrices x = ±a/e; its major axis is the line segment joining the points (±a, 0), and its minor axis is the line segment joining the points (0, ±b). See the following diagram.

42

LA2

A solution with at least one non-zero unknown is a non-trivial solution.

4 Elementary operations

The following operations do not change the solution set of a system of linear equations. 1. Interchange two equations. 2. Multiply an equation by a non-zero number. 3. Change one equation by adding to it a multiple

of another.

To solve a system of three linear equations in three unknowns, use elementary operations to try to

LA2 Linear equations and matrices

1 Simultaneous linear equations

1 Linear equations

An equation of the form

ax + by = c,

where a, b and c are real numbers, and a and b are not both zero, represents a line in R2 . There are

reduce the system to the form

= ∗⎧⎨ infinitely many solutions to this equation—one

a21x1 + a22x2 + · · · + a2nxn = b2,

corresponding to each point on the line. x , y = ∗An equation of the form ,

z = ∗. ⎩ax + by + cz = d,

where a, b, c and d are real numbers, and a, b and c are not all zero, represents a plane in R3 . There are infinitely many solutions to this equation—one corresponding to each point in the plane. An equation of the form

a1x1 + a2x2 + · · ·+ anxn = b,

where a1, a2, . . . , an, b are real numbers, and a1, . . . , an are not all zero, is a linear equation in the n unknowns x1, x2, . . . , xn. The numbers ai are the coefficients, and b is the constant term.

2 A system of m simultaneous linear equations in

⎧ ⎪⎪⎪⎨

2 Row-reduction

1 A matrix is simply a rectangular array of objects, usually numbers, enclosed in brackets. The objects in a matrix are called its entries. The entries along a horizontal line form a row, and those down a vertical line form a column.

2 The system

a11x1 + a12x2 + · · · + a1nxn = b1,

n unknowns, x1, . . . , xn, is as follows: ⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

. . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = bm,

a11x1 + a12x2 + · · · + a1nxn = b1, a21x1 + a22x2 + · · · + a2nxn = b2,

⎞⎛

. . .. . of m linear equations in n unknowns x1, x2, . . . , xn is . . . . . . . abbreviated as the augmented matrix ⎪⎪⎪⎩

am1x1 + am2x2 + · · · + amnxn = bm. a11 a12 · · · a1n b1

Such a system has a solution x1 = c1, x2 = c2, . . . , xn = cn, if these values simultaneously satisfy all m equations of the system. The solution set of the

⎜⎜⎜⎝

⎟⎟⎟⎠.

a21 a22 · · · a2n b2 . . . . . . . . . . . . m1 am2 · · · a bsystem is the set of all the solutions. a mn m

Any system of linear equations has a solution set which • contains exactly one solution,• or is empty,• or contains infinitely many solutions.A system of simultaneous linear equations is consistent when it has at least one solution, and inconsistent when it has no solutions.

3 A homogeneous system of linear equations is a system of simultaneous linear equations in which each constant term is zero. A system containing at least one non-zero constant term is a non-homogeneous system. The trivial solution (if this exists) to a system of simultaneous linear equations is the solution with each unknown equal to zero.

3 Elementary row operations are the following operations on the rows of the augmented matrix. 1. Interchange two rows. 2. Multiply a row by a non-zero number. 3. Change one row by adding to it a multiple of

another.

4 A matrix is in row-reduced form when: any zero rows are at the bottom;the leading entry of each non-zero row is a 1(called a leading 1);each leading 1 is to the right of the leading 1in the row above;each leading 1 is the only non-zero entry inits column.

43

LA2

3 Matrix addition

The sum of two m × n matrices A = (aij ) and B = (bij ) is the m × n matrix A + B = (aij + bij ).

Theorem 3.1 For all matrices A, B and C of the same size, A + B = B + A (commutative law), A + (B + C) = (A + B) + C (associative law).

The m × n zero matrix 0m,n is the m × n matrix in which all entries are 0.The zero matrix is the identity element for theoperation of matrix addition.The negative of an m × n matrix A = (aij ) is the m × n matrix

−A = (−aij ).

Theorem 3.2 Let A be a matrix. Then

A + (−A) = (−A) + A = 0.

The m × n matrix −A is the additive inverse of them × n matrix A.The set of m × n matrices under the operation ofmatrix addition forms a group.Using the negative of a matrix, we can subtractmatrices:

A −B = A + (−B).

4 The scalar multiple of an m × n matrix A = (aij ) by a scalar k is the m × n matrix kA = (kaij ).

Theorem 3.3 For all matrices A and B of the same size, and all scalars k, the distributive law holds; that is,

Strategy 2.1 Row-reducing a matrix. Carry out the following four steps, first with row 1 as the current row, then with row 2, and so on, until

either every row has been the current row, or step 1 is not possible.

1. Select the first column from the left that has at least one non-zero entry in or below the current row.

2. If the current row has a 0 in the selected column, interchange it with a row below it which has a non-zero entry in that column.

3. If the entry now in the current row and the selected column is c, multiply the current row by 1/c to create a leading 1.

4. Add suitable multiples of the current row to the others rows to make each entry above and below the leading 1 into a 0.

Strategy 2.2 To solve a system of linearequations by Gauss–Jordan elimination.1. Form the augmented matrix. 2. Obtain the row-reduced matrix. 3. Solve the simplified system of linear

equations.

3 Matrix algebra

1 A matrix of size m × n has m rows and n columns. An n × n matrix is called a square matrix. The entry in the ith row and jth column of a matrix A is called the (i, j)-entry, often denoted by aij . In general, we write A or (aij ) to denote a matrix: k(A + B) = kA + kB.⎞⎛

⎜⎜⎜⎝

· · · a1na11 a12

· · · a2na21 a22 . . . . . . . . .

⎟⎟⎟⎠A = = (aij ). 5 Matrix multiplication

The product of an m × n matrix A with an n × p matrix B is the m × p matrix AB whose (i, j)-entry is the dot product of the ith row of A with the jth

am1 am2 · · · amn

Two matrices A and B of the same size are equal if all their corresponding entries agree. We write column of B.A = B.

The product AB has the same number of rows as A, and the same number of columns as B. Matrix multiplication is associative; that is, the products (AB)C and A(BC) are equal (when they can be formed). The distributive law holds for multiplication of a matrix by a matrix; that is, A(B + C) = AB + AC, whenever these products can be formed.

2 We call a matrix with just one column a columnmatrix, and a matrix with just one row a row matrix.

44

LA2

6 Diagonal and triangular matrices

The entries of a square matrix from the top left-hand corner to the bottom right-hand corner are the diagonal entries; the diagonal entries form the main (or leading) diagonal of the matrix. For a square matrix A = (aij ) of size n× n, the diagonal entries are

a11, a22, . . . , ann.

A diagonal matrix is a square matrix each of whose non-diagonal entries is zero. Multiplication of diagonal matrices is commutative. A square matrix with each entry below the main diagonal equal to zero is an upper-triangular matrix. A square matrix with each entry above the main diagonal equal to zero is a lower-triangular matrix.

7 The identity matrix In is the n× n matrix in which each of the entries is 0 except those on the main diagonal, which are all 1.

Theorem 3.4 Let A be an m× n matrix. Then

ImA = AIn = A.

8 The transpose of an m× n matrix A is the n×m matrix AT whose (i, j)-entry is the (j, i)-entry of A.

Theorem 3.5 Let A and B be m× n matrices. Then the following results hold: (a) (AT )T = A; (b) (A + B)T = AT + BT . Let A be an m× n matrix and B an n× pmatrix. Then

⎧ ⎪⎪⎪⎨

(c) (AB)T = BT AT .

A square matrix A is symmetric if AT = A.

9 Any system of simultaneous linear equations a11x1 + a12x2 + · · · + a1nxn = b1,

4 Matrix inverses

1 Let A be a square matrix, and suppose that there exists a matrix B of the same size such that

AB = I and BA = I. Then B is an inverse of A.

Theorem 4.1 A square matrix has at most one inverse.

Theorem 4.2 A square matrix with a zero row has no inverse.

A square matrix that has an inverse is invertible. The unique inverse of an invertible matrix A is denoted by A−1 . For any invertible matrix A,

AA−1 = I and A−1A = I. If A is an invertible matrix, then A−1 is also invertible, with inverse A; that is,

(A−1)−1 = A.

Let A and B be invertible matrices of the same size. Then AB is invertible, and (AB)−1 = B−1A−1 .

Theorem 4.3 Let A1,A2, . . . ,Ak be invertible matrices of the same size. Then the product A1A2 · · ·Ak is invertible, with

= A−1A−1(A1A2 · · ·Ak )−1 · · ·A−1 .k k−1 1

Theorem 4.4 The set of all invertible n× n matrices forms a group under matrix multiplication.

Theorem 4.5 Invertibility Theorem (a) A square matrix is invertible if and only if

its row-reduced form is I. (b) Any sequence of elementary row operations

that transforms a matrix A to I also a21x1 + a22x2 + · · · + a2nxn = b2, transforms I to A−1 . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = bm,

⎪⎪⎪⎩

⎞⎛⎞⎛⎞⎛can be expressed in matrix form as Ax = b: a11 a12 · · · a1n x1 b1 ⎜⎜⎜⎝

⎜⎜⎜⎝

⎟⎟⎟⎠

⎟⎟⎟⎠

⎜⎜⎜⎝

⎟⎟⎟⎠

b2 ..

· · · a2na21 a22 . . .. . .

x2 .. = .

. . . . . am1 am2 · · · amn xn bm

A x b

A is called the coefficient matrix of the system.

45

( )

∣ ∣ ∣ ∣∣ ∣ ∣ ∣

LA2

Strategy 4.1 To determine whether or not a given square matrix A is invertible, and find its inverse if it is. Write down (A | I), and row-reduce it until the left half is in row-reduced form. • If the left half is the identity matrix, then

the right half is A−1 . • Otherwise, A is not invertible.

You may find it helpful to remember the following scheme:

(A | I) ↓

(I | A−1).

If it becomes clear while you are row-reducing (A | I) that the left half will not reduce to the identity matrix (for example, if a zero row appears in the left half), then you can stop the row-reduction and conclude that A is not invertible.

Theorem 4.6 Let A be an invertible matrix. Then the system of linear equations Ax = b has the unique solution x = A−1b.

Theorem 4.7 Let A be an n× n matrix. Then the following statements are equivalent. (a) A is invertible. (b) The system Ax = b has a unique solution

for each n× 1 matrix b. (c) The system Ax = 0 has only the trivial

solution.

Corollary Let E1,E2, . . . ,Ek be the m×m elementary matrices associated with a sequence of k elementary row operations carried out on a matrix A with m rows, in the same order. Then, after the sequence of row operations has been performed, the resulting matrix is

EkEk−1 · · ·E2E1A.

3 Given any elementary row operation, it is easy to write down an inverse elementary row operation that undoes the effect of the first, as summarised in the following table.

Elementary Inverse elementary row operation row operation

ri ↔ rj ri ↔ rj

ri → c ri (c = 0) ri → (1/c) ri

ri → ri + c rj ri → ri − c rj

Theorem 4.9 Let E1 and E2 be elementary matrices of the same size whose associated elementary row operations are inverses of each other. Then E1 and E2 are inverses of each other.

Corollary Every elementary matrix is invertible, and its inverse is also an elementary matrix.

5 Determinants

2 Elementary matrices 1 The determinant of a 2 × 2 matrix

A matrix obtained by performing an elementary row a bA = is

c doperation on an identity matrix is an elementary matrix.

det A =a b c d

= ad− bc.

Theorem 4.8 Let E be an elementary matrix, and let A be any matrix with the same number of rows as E. Then the product EA is the same as the matrix obtained when the elementary row operation associated with E is performed on A.

46

( )

( )

∣ ∣ ∣ ∣∣ ∣ ∣ ∣

∣ ∣ ∣ ∣∣ ∣ ∣ ∣

∣ ∣ ∣ ∣∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2

LA2

4 Properties of determinants

Strategy 5.1 To find the inverse of a 2 × 2 matrix Theorem 5.1 Let A and B be two square

matrices of the same size. Then the following hold: (a) det(AB) = (det A)(det B); (b) det I = 1;

(c) det AT = det A.

a bA =

c d

with det A = ad − bc = 0. 1. Interchange the diagonal entries. 2. Multiply the non-diagonal entries by −1. 3. Divide by the determinant, giving

1 d −b Theorem 5.2 Let E be an elementary .A−1 = ad − bc −c a matrix, and let k be a non-zero real number.

(a) If E results from interchanging two rows of I, then det E = −1.

(b) If E results from multiplying a row of Iby k, then det E = k.

The determinant of a 3 × 3 matrix a1 b1 c1

⎞⎛

A =⎝ ⎠a2 b2 c2 (c) If E results from adding k times one row of a3 b3 c3

I to another row, then det E = 1. is

det A = a1 − b1 a2 c2

a3 c3 + c1 .

b2 b2c2 a2

b3 b3 Two rows (or columns) of a matrix are proportional c3 a3

when one is a multiple of the other.

Theorem 5.3 Let A be a square matrix. Then det A = 0 if any of the following hold: (a) A has an entire row (or column) of zeros; (b) A has two equal rows (or columns); (c) A has two proportional rows (or columns).

3 A submatrix is a matrix formed from another matrix with some of the rows and/or columns removed. Let A = (aij ) be an n × n matrix. The cofactor Aij

associated with the entry aij is

Aij = (−1)i+j det Aij ,

where Aij is the (n − 1) × (n − 1) submatrix of A resulting when the ith row and jth column (the row and column containing the entry aij ) are covered up. The determinant of an n × n matrix A = (aij ) is

a12 a1n

a21 a22

∣∣ · · ·a11∣∣∣∣∣∣∣ · · · a2n

Theorem 5.4 A square matrix A isinvertible if and only if det A = 0.

det A = . . . . . . . . . an1 an2 · · · ann

Theorem 5.5 Let A and B be square = a11A11 + a12A12 + · · ·+ a1nA1n.

Strategy 5.2 To evaluate the determinant of an n × n matrix. 1. Expand along the top row to express the

n × n determinant in terms of n determinants of size (n − 1) × (n − 1).

2. Expand along the top row of each of the resulting determinants.

3. Repeatedly apply step 2 until the only determinants in the expression are of size 2 × 2.

4. Evaluate the final expression.

matrices of the same size. Then AB = I if and only if BA = I.

47

LA3

LA3 Vector spaces

1 Vector spaces

1 In R2, the set of ordered pairs of real numbers, the operations of addition and multiplication by a scalar are defined as follows:

(u1, u2) + (v1, v2) = (u1 + v1, u2 + v2); α(u1, u2) = (αu1, αu2), where α ∈ R.

In R3, the set of ordered triples of real numbers, the operations of addition and multiplication by a scalar are defined as follows:

(u1, u2, u3) + (v1, v2, v3) = (u1 + v1, u2 + v2, u3 + v3); α(u1, u2, u3) = (αu1, αu2, αu3), where α ∈ R.

2 If n is a positive integer, then an ordered n-tuple is a sequence of real numbers (u1, u2, . . . , un). The set of all ordered n-tuples is called n-dimensional space, and is denoted by Rn .

3 A real vector space consists of a set V of elements and two operations, vector addition and scalar multiplication, such that the following axioms hold.

A1 closure For all v1,v2 ∈ V , v1 + v2 ∈ V.

A2 identity For each v ∈ V , there is a zero element 0 ∈ V satisfying

v + 0 = 0 + v = v.

A3 inverses For each v ∈ V , there is an element −v (its additive inverse) such that

v + (−v) = (−v) + v = 0.

A4 associativity For all v1,v2,v3 ∈ V , (v1 + v2) + v3 = v1 + (v2 + v3).

A5 commutativity For all v1,v2 ∈ V , v1 + v2 = v2 + v1.

S1 closure For all v ∈ V and α ∈ R, αv ∈ V.

S2 associativity For all v ∈ V and α, β ∈ R, α(βv) = (αβ)v.

S3 identity For all v ∈ V , 1v = v.

D1 distributivity For all v1,v2 ∈ V and α ∈ R, α(v1 + v2) = αv1 + αv2.

D2 distributivity For all v ∈ V and α, β ∈ R, (α + β)v = αv + βv.

Examples of real vector spaces

• R, R2 , R3, and (more generally) Rn (n ≥ 1). • V = a cos x + b sin x : a, b ∈ R. • Pn, the set of all real polynomials of degree less

than n. • C, the set of complex numbers. • The set Mm,n of all m× n matrices with real

entries. • R∞, the set of all infinite sequences of real

numbers.

2 Linear combinations and spanning sets

1 Let v1,v2, . . . ,vk belong to a vector space V . Then a linear combination of the vectors v1,v2, . . . ,vk is a vector of the form

α1v1 + α2v2 + · · ·+ αk vk,

where α1, α2, . . . , αk are real numbers. This vector also belongs to V .

Strategy 2.1 To determine whether a given vector v can be written as a linear combination of the vectors v1,v2, . . . ,vk. 1. Write v = α1v1 + α2v2 + · · ·+ αk vk . 2. Use this expression to write down a system

of simultaneous linear equations in the unknowns α1, α2, . . . , αk.

3. Solve the resulting system of equations, if possible.

Then v can be written as a linear combination of v1,v2, . . . ,vk if and only if the system has a solution.

2 Let S = v1,v2, . . . ,vk be a finite set of vectors in a vector space V . Then the span 〈S〉 of S is the set of all possible linear combinations

α1v1 + α2v2 + · · ·+ αk vk,

where α1, α2, . . . , αk are real numbers; that is, 〈S〉 = α1v1 + α2v2 + · · ·+ αk vk : α1, α2, . . . , αk ∈ R. We say that the set of vectors v1,v2, . . . ,vk spans 〈S〉 or is a spanning set for 〈S〉, and that 〈S〉 is the set spanned by S.

3 Bases and dimension

1 A minimal spanning set of a vector space V is a set containing the smallest number of vectors that span V .

48

LA3

Theorem 3.1 Suppose that the vector vk

can be written as a linear combination of the vectors v1, . . . ,vk−1. Then the span of the set v1,v2, . . . ,vk is the same as the span of the set v1,v2, . . . ,vk−1.

2 A finite set of vectors v1,v2, . . . ,vk in a vector space V is linearly dependent if there exist real numbers α1, α2, . . . , αk , not all zero, such that

α1v1 + α2v2 + · · ·+ αk vk = 0.

A finite set of vectors v1,v2, . . . ,vk is linearly independent if it is not linearly dependent; that is, if

α1v1 + α2v2 + · · ·+ αk vk = 0

only when α1 = α2 = · · · = αk = 0.

A linearly independent set cannot contain the zero vector.Any set consisting of just one non-zero vector islinearly independent.Any set of two non-zero vectors is linearly dependentif one of the vectors is a multiple of the other, andlinearly independent otherwise.Any set of three non-zero vectors is linearlyindependent if and only if the vectors are notcoplanar.

Strategy 3.1 To test whether a given set of vectors v1,v2, . . . ,vk is linearly independent. 1. Write down the equation

α1v1 + α2v2 + · · ·+ αk vk = 0.2. Express this equation as a system of

simultaneous linear equations in theunknowns α1, α2, . . . , αk .

3. Solve these equations.If the only solution is α1 = α2 = · · · = αk = 0, then the set of vectors is linearly independent.If there is a solution with at least one ofα1, α2, . . . , αk not equal to zero, then the set ofvectors is linearly dependent.

3 A basis for a vector space V is a linearly independent set of vectors which is a spanning set for V .

Theorem 3.2 Let S be a basis for a vector space V . Then each vector in V can be expressed as a linear combination of the vectors in S in only one way.

Strategy 3.2 To determine whether a set of vectors S in a vector space V is a basis for V , check the following conditions. (1) S is linearly independent.(2) S spans V .If both (1) and (2) hold, then S is a basis for V .If either (1) or (2) does not hold, then S is nota basis for V .

4 Standard bases

For R2, the standard basis is (1, 0), (0, 1).

For R3, the standard basis is (1, 0, 0), (0, 1, 0), (0, 0, 1).

For Rn, the standard basis is the set of n vectors (1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, 0, . . . , 1).

For Pn, the standard basis is 2 n−11, x, x , . . . , x .

For M2,2, the standard basis is ( ) ( ) ( ) ( ) 1 0 0 1 0 0 0 0 0 0

, 0 0 , 1 0

, 0 1 .

For C, the standard basis is 1, i.

5 Let E = e1, e2, . . . , en be a basis for a vector space V , and suppose that

v = v1e1 + v2e2 + · · ·+ vnen,

where v1, . . . , vn ∈ R. Then the E-coordinate representation of v is

vE = (v1, v2, . . . , vn)E .

We call v1, . . . , vn the coordinates of v with respect to the basis E, or, more briefly, the E-coordinates of v. If E is the standard basis, then we refer to the standard coordinate representation, standard coordinates, and so on; in this case, the subscript E is usually omitted.

6 Let V be a vector space. Then V is finite-dimensional if it contains a finite set of vectors S which forms a basis for V . If no such set exists, then V is infinite-dimensional.

Theorem 3.3 Let E = e1, e2, . . . , en be a basis for a vector space V , and let S = v1,v2, . . . ,vm be a set of m vectors in V , where m > n. Then S is a linearly dependent set.

49

LA3

Corollary Let V be a vector space with a basis containing n vectors. If a linearly independent subset of V contains m vectors, then m ≤ n.

Theorem 3.4 Basis Theorem Let V be a finite-dimensional vector space. Then every basis for V contains the same number of vectors.

The dimension of a finite-dimensional vector space V , denoted by dim V , is the number of vectors in any basis for the space.

Theorem 3.5 Let V be an n-dimensional vector space. Then any set of n linearly independent vectors in V is a basis for V .

Strategy 3.3 To determine whether a set of vectors S in Rn is a basis for the vector space Rn, check the following conditions. (1) S contains n vectors. (2) S is linearly independent.If both (1) and (2) hold, then S is a basisfor Rn .If either (1) or (2) does not hold, then S is not a basis for Rn .

Theorem 3.6 Let S = v1,v2, . . . ,vm be a linearly independent subset of an n-dimensional vector space V , where m < n. Then there exist vectors vm+1, . . . ,vn in V such that v1,v2, . . . ,vn is a basis for V .

4 Subspaces

A subset S of a vector space V is a subspace of V if S is itself a vector space under vector addition and scalar multiplication as defined in V .

Theorem 4.1 A subset S of a vector space V is a subspace of V if it satisfies the following conditions.

(a) 0 ∈ S.

(b) S is closed under vector addition.

(c) S is closed under scalar multiplication.

Strategy 4.1 To test whether a given subset S of a vector space V is a subspace of V , check the following conditions.

(1) 0 ∈ S (zero vector).

(2) If v1,v2 ∈ S, then v1 + v2 ∈ S (vectoraddition).

(3) If v ∈ S and α ∈ R, then αv ∈ S (scalarmultiplication).

If (1), (2) and (3) hold, then S is a subspace of V .If any of (1), (2) or (3) does not hold, then S isnot a subspace of V .

Theorem 4.2 Let S be a non-empty finite subset of a vector space V . Then 〈S〉 is a subspace of V .

Theorem 4.3 The dimension of a subspace of a vector space V is less than or equal to the dimension of V .

5 Orthogonal bases

1 Let v = (v1, v2, . . . , vn) and w = (w1, w2, . . . , wn) be vectors in Rn. The dot product of v and w is the real number

v . w = v1w1 + v2w2 + · · ·+ vnwn.

2 The vectors v and w in Rn are orthogonal if v . w = 0.

A set of vectors in Rn is an orthogonal set if any two distinct vectors in the set are orthogonal. An orthogonal basis for Rn is an orthogonal set which is a basis for Rn .

Theorem 5.1 Let S = v1,v2, . . . ,vk be an orthogonal set of non-zero vectors in Rn . Then S is a linearly independent set.

50

( ) ( )

( )

( )

( ) ( )

. ( ) ( )

( )

√

Theorem 5.2 Any orthogonal set of n non-zero vectors in Rn is an orthogonal basis for Rn .

3 Finding bases

Strategy 5.1 To find an orthogonal basis for R3 containing a given vector v1. Find a plane which is orthogonal to the given vector, and then find an orthogonal basis v2,v3 for the plane. Then v1,v2,v3 is a suitable basis.

A hyperplane is a three-dimensional subspace of R4 .

LA3

5 Changing a basis in Rn to an orthogonal basis in Rn

Theorem 5.4 Gram–Schmidt orthogonalisation process Let w1,w2, . . . ,wn be a basis for Rn, and let

v1 = w1, v1 . w2 v2 = w2 − v1, v1 . v1

v1 . w3 v2 . w3 v3 = w3 − v1 − v2, v1 . v1 v2 . v2 . . v1 . wn v2 . wn vn = wn − v1 − v2 v1 . v1 v2 . v2

vn−1 . wn− · · · − vn−1. vn−1 . vn−1

Then v1,v2, . . . ,vn is an orthogonal basis for Rn .Strategy 5.2 To find an orthogonal basis

for R4 containing a given vector v1. Find a hyperplane which is orthogonal to the given vector, and then find an orthogonal basis v2,v3,v4 for the hyperplane. Then v1,v2,v3,v4 is a suitable basis.

4 Expressing vectors in terms of orthogonal bases

Theorem 5.3 Let v1,v2, . . . ,vn be an orthogonal basis for Rn and let u be any vector in Rn. Then

. u . uv1 v2 u = v1 + v2 + · · · v1 . v1 v2 . v2

vn . u + vn.

vn . vn

Strategy 5.3 To express a vector u in terms of an orthogonal basis v1,v2, . . . ,vn.

v2 . u1. Calculate α1 = v1 . u

, α2 = , . . . , v1 . v1 v2 . v2 vn . u

αn = .vn . vn

2. Write u = α1v1 + α2v2 + · · ·+ αnvn.

(This process can also be used for a subspace of dimension n of Rm , m ≥ n.)

6 Let v = (v1, v2, . . . , vn) be a vector in Rn. Then the length of v is

√ 2 2‖v‖ = v . v = v2 + v2 + · · ·+ v .1 n

7 An orthonormal basis for Rn is an orthogonal basis in which each basis vector has length 1.

Strategy 5.4 To construct an orthonormal basis for Rn from an orthogonal basis v1,v2, . . . ,vn for Rn . 1. Calculate the length of each basis vector. 2. Divide each basis vector by its length. The required orthonormal basis is

v1 v2 vn , , . . . , . ‖v1‖ ‖v2‖ ‖vn‖

Theorem 5.5 Let v1,v2, . . . ,vn be an orthonormal basis for Rn, and let u be any vector in Rn. Then

u = (v1 . u)v1 + (v2 . u)v2 + · · ·+ (vn . u)vn.

51

( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

) ( ) ( )

( )

LA4

LA4 Linear transformations

1 Introducing linear transformations

1 The following are linear transformations. A k-dilation of R2 stretches (or scales) vectors radially from the origin by a factor k, where k is any real number. This can be represented by

x k 0 x kx −→ = y

0 k y ky.

A (k, l)-stretching of R2 stretches (or scales) vectors by a factor k in the x-direction and by a factor l in the y-direction, where k and l are any real numbers. This can be represented by

x k 0 x kx −→ = y

0 l y ly.

A rotation rθ of R2 rotates vectors anticlockwise through an angle θ about the origin (0, 0). This can be represented by

x cos θ − sin θ x −→ y sin θ cos θ y

x cos θ− y sin θ = x sin θ + y cos θ .

A reflection qφ of R2 reflects vectors in the straight line through the origin that makes an angle φ with the (

x-axis. This can be represented by

x cos 2φ sin 2φ x −→ y sin 2φ − cos 2φ y

x cos 2φ + y sin 2φ = x sin 2φ− y cos 2φ

.

2 Let V and W be vector spaces. A function t : V −→ W is a linear transformation if itsatisfies the following properties.LT1 t(v1 + v2) = t(v1) + t(v2), for all v1,v2 ∈ V .LT2 t(αv) = α t(v), for all v ∈ V , α ∈ R.(Throughout this unit, V and W denote vectorspaces.)

Theorem 1.1 Let t : V −→ W be a linear transformation. Then t(0) = 0.

Strategy 1.1 To determine whether or not a given function t : V −→ W is a linear transformation. 1. Check whether t(0) = 0; if not, then t is not

a linear transformation. 2. Check whether t satisfies the properties LT1

and LT2. The function t is a linear transformation if and only if both these properties are satisfied.

If either of LT1 or LT2 fails, then you do not need to check the other.

3 A shear of R2 in the x-direction by a factor k is the linear transformation

t : R2 −→ R2

(x, y) −→ (x + ky, y). A translation of R2 by (a, b) is the function

t : R2 −→ R2

(x, y) −→ (x + a, y + b).

4 The zero transformation from V to W is the linear transformation

t : V −→ W

v −→ 0.

The identity transformation of V is the linear transformation

iV : V −→ V

v −→ v.

We omit the subscript V when the vector space is clear from the context.

5 Linear combinations of vectors

Theorem 1.2 A function t : V −→ W is a linear transformation if and only if it satisfies LT3 t(α1v1 + α2v2) = α1t(v1) + α2t(v2),

for all v1,v2 ∈ V and all α1, α2 ∈ R.

Theorem 1.3 Let t : V −→ W be a linear transformation. Then

t(α1v1 + α2v2 + · · · + αnvn) = α1t(v1) + α2t(v2) + · · · + αnt(vn),

for all v1, . . . ,vn ∈ V and all α1, . . . , αn ∈ R, n ∈ N.

52

2 Matrices of linear transformations

1 Let V and W be vector spaces of dimensions n and m, respectively. Let t : V −→ W be a linear transformation, let E = e1, . . . , en be a basis for V , let F = f1, . . . , fm be a basis for W , and let A be an m× n matrix such that

t(v)F = AvE , for each vector v in V.

Then vE −→ AvE = t(v)F is the matrix representation of t with respect to the bases E and F , and A is the matrix of t with respect to the bases E and F .

Theorem 2.1 Let t : V −→ W be a linear transformation, let E = e1, . . . , en be a basis for V and let F = f1, . . . , fm be a basis for W . Let

t(e1) = (a11, a21, . . . , am1)F ,

t(e2) = (a12, a22, . . . , am2)F ,.. .

t(en) = (a1n, a2n, . . . , amn)F .

Then there is exactly one matrix of t with

LA4

2 A linear transformation t : V −→ W has many different matrix representations:

different bases for V and W give different matrix representations.

Moreover, the order of the elements in a basis is important:

a different order gives a different matrix representation.

Theorem 2.2 Let t : V −→ W be a function which has a matrix representation. Then t is a linear transformation.

The linear transformations from a finite-dimensional vector space V to a finite-dimensional vector space W are precisely those functions from V to W that have a matrix representation.

3 Composition and invertibility

1 Combining linear transformations

respect to the bases E and F , namely

a12 · · · a1n Composition Rule ⎞

Theorem 3.1 a11 ⎛

A =⎜⎜⎜⎝

· · · a2na21 a22 . . .. . . . . .

⎟⎟⎟⎠. Let t : V −→ W and s : W −→ X be linear

transformations. Then:

⎞

· · · amnam1 am2

Strategy 2.1 To find the matrix A of t with

⎛

respect to the bases E and F .

1. Find t(e1), t(e2), . . . , t(en). 2. Find the F -coordinates of each of these

image vectors: t(e1) = (a11, a21, . . . , am1)F ,

t(e2) = (a12, a22, . . . , am2)F , . . .

t(en) = (a1n, a2n, . . . , amn)F .

3. Construct the matrix A column by column:

A = ⎜⎜⎜⎝

· · · a1na11 a12

· · · a2na21 a22 . . . . . . . . .

· · · amnam1 am2

⎟⎟⎟⎠.

(a) s t : V −→ X is a linear transformation; (b) if A is the matrix of t with respect to the

bases E and F , and B is the matrix of s with respect to the bases F and G, then BA is the matrix of s t with respect to the bases E and G.

53

LA4

Corollary Let A, B and C be matrices of sizes q × p, p× m and m× n, respectively. Then

A(BC) = (AB)C.

2 The linear transformation t : V −→ W is invertible if there exists an inverse function t−1 : W −→ V such that

−1 −1t t = iV and t t = iW .

Theorem 3.2 Inverse Rule Let t : V −→ W be a linear transformation. (a) If t is invertible, then t−1 : W −→ V is also

a linear transformation. (b) If A is the matrix of t with respect to the

bases E and F , then: (i) t is invertible if and only if A is invertible; (ii) if t is invertible, then A−1 is the matrix of t−1 with respect to the bases F and E.

Corollary Let t : V −→ W be an invertible linear transformation, where V and W are finite dimensional. Then

dim V = dim W.

Strategy 3.1 To determine whether or not a linear transformation t : V −→ W is invertible, where V and W are n-dimensional vector spaces with bases E and F , respectively. 1. Find a matrix representation of t,

vE −→ AvE = t(v)F .

2. Evaluate det A.

If det A = 0, then t is not invertible.If det A = 0, then t is invertible and t−1 : W −→ V has the matrix representation

wF −→ A−1wF = t−1(w)E .

3 The vector spaces V and W are isomorphic if there exists an invertible linear transformation t : V −→ W . Such a function t is an isomorphism.

Theorem 3.3 The finite-dimensional vector spaces V and W are isomorphic if and only if

dim V = dim W.

4 Image and kernel

1 The image of a linear transformation t : V −→ W is the set

Im(t) = w ∈ W : w = t(v), for some v ∈ V .

Theorem 4.1 Let t : V −→ W be a linear transformation. Then Im(t) is a subspace of the codomain W .

Strategy 4.1 To find a basis for Im(t), where t : V −→ W is a linear transformation. 1. Find a basis e1, . . . , en for the domain V . 2. Determine the vectors t(e1), . . . , t(en). 3. If there is a vector v in

S = t(e1), . . . , t(en) that is a linear combination of the other vectors in S, then discard v to give the set S1 = S − v .

4. If there is a vector v1 in S1 such that v1 is a linear combination of the other vectors in S1, then discard v1 to give the set S2 = S1 − v1 .

Continue discarding vectors in this way until you obtain a linearly independent set. This set is a basis for Im(t).

A linear transformation t : V −→ W is onto if and only if Im(t) = W .

54

LA4

2 The kernel of a linear transformation t : V −→ W is the set

Ker(t) = v ∈ V : t(v) = 0 .

Theorem 4.2 Let t : V −→ W be a linear transformation. Then Ker(t) is a subspace of the domain V .

Theorem 4.3 Solution Set Theorem Let t : V −→ W be a linear transformation. Let b ∈ W and let a be one vector in V that maps to b, that is, t(a) = b. Then the solution set of the equation t(x) = b is x : x = a + k for some k ∈ Ker(t) .

A linear transformation t : V −→ W is one-one if and only if Ker(t) = 0 . 3 Dimensions

Theorem 4.4 Dimension Theorem Let t : V −→ W be a linear transformation.Then

dim Im(t) + dim Ker(t) = dim V.

Theorem 4.5 Let t : V −→ W be a linear transformation from an n-dimensional vector space V to an m-dimensional vector space W . (a) If n > m, then t is not one-one, since

Ker(t) = 0 . (b) If n < m, then t is not onto, since

Im(t) = W . (c) If n = m, then

either t is both one-one and onto, since Ker(t) = 0 and Im(t) = W ;

or t is neither one-one nor onto, since = 0 and Im(t) Ker(t) = W .

4 Number of solutions of a system of linear equations

Theorem 4.6 Let Ax = b be a system of m simultaneous linear equations in n unknowns. (a) If n > m, then Ax = b has either no

solutions or infinitely many solutions.(b) If n < m, then there is some b for which

Ax = b has no solutions. (c) If n = m, then:

either Ax = b has exactly one solution for each b;

or there are some b for which Ax = b has no solutions, and for all other b, Ax = b has infinitely many solutions.

55

LA5

LA5 Eigenvectors

1 Eigenvalues and eigenvectors

1 Let t : V −→ V be a linear transformation. An eigenvector of t is a non-zero vector v that is mapped by t to a scalar multiple of itself; this scalar is the corresponding eigenvalue. In symbols, a non-zero vector v is an eigenvector of a linear transformation t if

t(v) = λv, for some λ ∈ R; λ is the corresponding eigenvalue.We exclude the case v = 0, since t(0) = 0 for everylinear transformation t.

2 A non-zero vector v is an eigenvector of asquare matrix A if

Av = λv, for some λ ∈ R; λ is the corresponding eigenvalue. The characteristic equation of a square matrix A is the equation

det(A − λI) = 0. The matrix A − λI is obtained by subtracting λ fromeach entry on the diagonal of A.The equations (A − λI)v = 0 are the eigenvectorequations.

Strategy 1.1 To determine the eigenvalues and eigenvectors of a square matrix A. 1. Find the eigenvalues.

Write down the characteristic equation

det(A − λI) = 0.Expand this determinant to obtain apolynomial equation in λ.Solve this equation to find the eigenvalues.

2. Find the eigenvectors. Write down the eigenvector equations

(A − λI)v = 0.

3 Eigenspaces

Theorem 1.2 Let t : V −→ V be a linear transformation. For each eigenvalue λ of t, let S(λ) be the set of vectors satisfying t(v) = λv. Then S(λ) is a subspace of V .

S(λ) is the set of eigenvectors corresponding to λ, together with the zero vector 0. It is called the eigenspace of t corresponding to the eigenvalue λ.

4 If the characteristic equation of a square matrix A can be written as

(λ− λ1)m1 (λ− λ2)m2 . . . (λ− λp)mp = 0, where λ1, λ2, . . . , λp are distinct, then the eigenvalue λj of A has multiplicity mj , for j = 1, 2, . . . , p. The dimension of an eigenspace cannot exceed the multiplicity of the corresponding eigenvalue.

2 Diagonalising matrices

1 Let t : Rn −→ Rn be a linear transformation, and let E be a basis for Rn consisting of eigenvectors of t. The basis E is an eigenvector basis of t.

Strategy 2.1 To find the matrix A of a linear transformation t : V −→ V with respect to the basis E = e1, e2, . . . , en. 1. Find the images t(e1), t(e2), . . . , t(en). 2. Find the E-coordinates of the image vectors

from step 1. 3. For each j = 1, 2, . . . , n, use the

E-coordinates of t(ej ) to form column j of the matrix A.

Theorem 2.1 Let t : Rn −→ Rn be a linear transformation, let E = e1, e2, . . . , en be an eigenvector basis of t, and let t(ej ) = λj ej , for j = 1, 2, . . . , n. Then the matrix of t with

For each eigenvalue λ, solve this system of linear equations to find the corresponding

⎞⎛respect to the eigenvector basis E is λ1 0 · · · 0

eigenvectors. ⎜⎜⎝0 λ2 · · · 0 . . .. . . . . . 0 0 · · · λn

⎟⎟⎠.D =

The sum of the eigenvalues equals the sum of the diagonal entries of the matrix A (the trace of A).

2 Let E = e1, e2, . . . , en be a basis for Rn. The transition matrix P from the basis E to the Theorem 1.1 The eigenvalues of a

triangular matrix and of a diagonal matrix arethe diagonal entries of the matrix.

standard basis for Rn is the matrix whose jth column is formed from the standard coordinates of ej .

56

LA5

Strategy 2.2 To diagonalise an n× n matrix A. 1. Find all the eigenvalues of A. 2. Find (if possible) an eigenvector basis

E = e1, . . . , en of A. 3. Write down the transition matrix P whose

jth column is formed from the standard coordinates of ej .

4. Then

Theorem 2.2 Let E = e1, e2, . . . , en be a basis for Rn, and let P be the transition matrix from the basis E to the standard basis for Rn . Then the standard coordinate representation of a vector in Rn is given by

v = PvE .

Moreover, P is invertible and

vE = P−1v. ⎛ ⎞λ1 0 · · · 0

When E is the standard basis for Rn, the transition matrix P is the identity matrix In.

⎜⎜⎝0 λ2 · · · 0 . . .. . . . . .

⎟⎟⎠P−1AP = D = ,

The rows or columns of an n× n matrix A form a set of n linearly independent vectors if and only if det A = 0.

Theorem 2.3 Let t : Rn −→ Rn be a linear transformation, and let E be an eigenvector basis of t. Let A be the matrix of t with respect to the standard basis for Rn, let D be the matrix of t with respect to the eigenvector basis E, and let P be the transition matrix from E to the standard basis for Rn. Then

D = P−1AP.

0 0 · · · λn

where λj is the eigenvalue corresponding to the eigenvector ej .

Theorem 2.4 Let A be an n× n matrix with distinct eigenvalues λ1, λ2, . . . , λn and corresponding eigenvectors e1, e2, . . . , en. Then E = e1, e2, . . . , en is an eigenvector basis of A.

Strategy 2.3 To find an eigenvector basis of an n× n matrix A. 1. Find a basis for each eigenspace of A. 2. Form the set E of all the basis vectors

found in step 1. If there are n vectors in E, then E is an eigenvector basis of A; otherwise E is not a basis.

3 Symmetric matrices

1 An orthonormal basis consists of mutually perpendicular (orthogonal) vectors of unit length. An n× n matrix P whose columns form an orthonormal basis for Rn is an orthogonal matrix; in this case we have P−1 = PT .

2 The matrix A is orthogonally diagonalisable if there exists an orthogonal matrix P such that the matrix

D = PT AP = P−1AP

is diagonal. If A is orthogonally diagonalisable, then it is symmetric.

3 The matrix A is diagonalisable if there exists an invertible matrix P such that the matrix

D = P−1AP

is diagonal. If D = P−1AP, then

PDnP−1 = An , for n = 1, 2, . . . .

4 Let A be an n× n matrix, and let E = e1, . . . , en be a basis for Rn consisting of eigenvectors of A. The basis E is an eigenvector basis of A.

57

( )

( )

( )

( )

′

( )

LA5

Strategy 3.1 To orthogonally diagonalise an n × n symmetric matrix A. 1. Find all the eigenvalues of A. 2. Find an orthonormal eigenvector basis

E = e1, e2, . . . , en of A.3. Write down the orthogonal transition

matrix P whose jth column is formed from the standard coordinates of ej .

4. Then ⎛ ⎞

Linear transformations of R3 whose matrices are orthogonal are rotations about a line through the origin, reflections in a plane through the origin, or combinations of these. The orthogonal matrices representing rotations of R3 are precisely those with determinant +1.

4 Conics and quadrics

1 The three types of non-degenerate conic are λ1 0 · · · 0 shown on page 101. 0 λ2 · · · 0 . . . . . . . . .

PT AP = D = ⎜⎜⎝

⎟⎟⎠, Strategy 4.1 To write the conic with equation

Ax2 + Bxy + Cy2 + Fx + Gy + H = 0

0 0 · · · λn

where λj is the eigenvalue corresponding to the eigenvector ej .

in standard form.

1. Introduce matrices. A 1 B2 1 B C2

Theorem 3.1 Eigenvectors corresponding to Write down the matrices A =

distinct eigenvalues of a symmetric matrix are F .and J = orthogonal. G

2. Align the axes. (a) Orthogonally diagonalise A:

Strategy 3.2 To find an orthonormal PT AP = λ1 0

.eigenvector basis of an n × n symmetric 0 λ2

matrix A. = JT P, and write the (b) Find f g 1. Find an orthonormal basis for each

eigenspace of A.2. Form the set E of all the basis vectors

found in step 1. This is the required basis.

3 Orthogonal matrices

conic in the form

λ1(x ′)2 + λ2(y ′)2 + fx′ + gy

+ H = 0. (∗) 3. Translate the origin.

Complete the squares in equation (∗), and change to the coordinate system (x′′, y′′). If λ1, λ2 = 0, then

Theorem 3.2 A square matrix P is ′′ g(x , y ′′) = x ′ +

f , y ′ +orthogonal if and only if PT = P−1 . .

2λ1 2λ2

2 A quadric in R3 is the set of points (x, y, z) that satisfy an equation of the form

Ax2 + By2 + Cz2 + Fxy + Gyz + Hxz

+ Jx + Ky + Lz + M = 0, where A, B, C, F , G and H are not all 0. The six types of non-degenerate quadric are shown on page 101.

Corollary Let P and Q be orthogonal n × n matrices. Then: (a) P−1(= PT ) is orthogonal; (b) the rows of P form an orthonormal basis

for Rn; (c) det P = ±1; (d) the product PQ is orthogonal.

4 Linear transformations of R2 whose matrices are orthogonal are rotations about the origin when the determinant is +1, and reflections in a line through the origin when the the determinant is −1.

58

( )

′′

′′ ′′ ( )

LA5

Strategy 4.2 To write the quadric with equation

Ax2 + By2 + Cz2 + Fxy + Gyz + Hxz

+ Jx + Ky + Lz + M = 0

in standard form.

1. Introduce matrices. Write down the matrices ⎛

A 1 1 ⎞ ⎛ ⎞

F H J2 2 ⎜ 1 F B 1 G ⎠ , J = ⎝ K ⎠.A = ⎝ 2 ⎟

21H 1 G C L 2 2

2. Align the axes. (a) Orthogonally diagonalise A: ⎛ ⎞

λ1 0 0PT AP = ⎝ 0 λ2 0 ⎠ .

0 0 λ3

(b) Find f g h = JT P, and write the quadric in the form

λ1(x ′)2 + λ2(y ′)2 + λ3(z ′)2

+ fx′ + gy ′ + hz′ + M = 0. (∗) 3. Translate the origin.

Complete the squares in equation (∗), and change to the coordinate system (x , y′′, z′′). If λ1, λ2, λ3 = 0, then

(x , y , z ′′) =

g x ′ +

2f

λ1 , y ′ + , z ′ +

h.

2λ2 2λ3

59

Analysis Block A

AA1

AA1 Numbers

1 Real numbers

1 The set of natural numbers is the set N = 1, 2, 3, . . .;

the set of integers is the set Z = . . . ,−2,−1, 0, 1, 2, . . .;

the set of rational numbers consists of all fractions, Q = p/q : p ∈ Z, q ∈ N.

These numbers can all be represented geometrically as points on a number line.The rationals have a natural order on the numberline: if a lies to the left of b on the number line, then

a is less than b or b is greater than a

and we write these as strict inequalities, a < b or b > a.

We write the weak inequalities, a ≤ b or b ≥ a, if either a < b or a = b.2 A decimal is an expression of the form

± a0.a1a2a3 . . . ,

where a0 is a non-negative integer, and a1, a2, a3, . . . are digits (i.e. numbers from the set 0, 1, 2, . . . , 9). If only a finite number of the digits a1, a2, . . . are non-zero, then the decimal is called terminating or finite; otherwise, we have a non-terminating or infinite decimal. A recurring decimal is a decimal with a recurring block of digits; for example, 0.863 63 . . . is written as 0.863. By definition, 0.9 = 1.

We order two rational numbers by examining theirdecimal representations and noticing the first placeat which the digits differ.Every rational number can be represented by a finiteor recurring decimal.3 A number which is not rational is calledirrational.The set of irrational numbers consists of all thenon-recurring decimals.

4 Together, the rational numbers and irrationalnumbers form the set of real numbers, denotedby R.We order two real numbers by examining theirdecimal representations and noticing the first placeat which the digits differ.The number line, complete with both rational andirrational points, is called the real line.

There is a one-one correspondence between the points on the real line and the set R of real numbers. We often use the word ‘point’ to mean ‘number’ in this context.

5 Order properties of R

Trichotomy Property If a, b ∈ R, then exactly one of the following holds:

a < b or a = b or a > b.

Transitive Property If a, b, c ∈ R, then

a < b and b < c ⇒ a < c.

Archimedean Property If a ∈ R, then there is a positive integer n such that

n > a.

Density Property If a, b ∈ R and a < b, then there is a rational number x and an irrational number y such that

a < x < b and a < y < b.

6 Arithmetic in R

Addition A1 If a, b ∈ R, then a + b ∈ R. closure A2 If a ∈ R, then

a + 0 = 0 + a = a. identity A3 If a ∈ R, then there is a number

−a ∈ R such that a + (−a) = (−a) + a = 0. inverses

A4 If a, b, c ∈ R, then (a + b) + c = a + (b + c). associativity

A5 If a, b ∈ R, then a + b = b + a. commutativity

Multiplication M1 If a, b ∈ R, then a× b ∈ R. closure M2 If a ∈ R, then

a× 1 = 1 × a = a. identity M3 If a ∈ R − 0, then there is a

number a−1 ∈ R such that a× a−1 = a−1 × a = 1. inverses

M4 If a, b, c ∈ R, then (a× b) × c = a× (b× c). associativity

M5 If a, b ∈ R, then a× b = b× a. commutativity

Addition and multiplication D If a, b, c ∈ R, then

a× (b + c) = a× b + a× c. distributivity

Any system satisfying the properties listed above is called a field.

60

( )

∑ ( )

∑ ( )

(

√

AA1

2 Inequalities 3 Proving inequalities

1 Rules for inequalities 1 Transitive Rule a < b and b < c ⇒ a < c. (1) a < b ⇔ b− a > 0. (2) a < b ⇔ a + c < b + c. (3) If c > 0, then

a < b ⇔ ac < bc; if c < 0, then

Combination Rules If a < b and c < d, then Sum Rule a + c < b + d; Product Rule ac < bd, provided a, c ≥ 0.

There are also versions of these rules involving weak inequalities.

a < b ⇔ ac > bc.

(4) If a, b > 0, then

a < b ⇔ 1 1

> a b

.

(5) If a, b ≥ 0 and p > 0, then

Triangle Inequality If a, b ∈ R, then

1. |a + b| ≤ |a|+ |b| (usual form); |a− b| ≥

∣∣ | − |b ∣∣2. (‘backwards’ form). | |a

a < b ⇔ ap < bp. There is a more general form of inequality 1:

There are corresponding versions of Rules 1–5 in which the strict inequality a < b is replaced by the if a1, a2, . . . , an ∈ R, then

weak inequality a ≤ b. |a1 + a2 + · · ·+ an| ≤ |a1|+ |a2|+ · · ·+ |an|. We frequently use the usual rules for the sign of a 2product:

The notation n!n = × + − k k! (n− k)!

is also denoted by nCk.+ + − − − +

In particular, the square of any real number is Theorem 3.1 Binomial Theorem non-negative. 1. If x ∈ R and n ∈ N, then

n2 The solution set of an inequality involving an n kunknown real number x is the set of values of x for (1 + x)n = x k

which the given inequality holds. k=0

To solve the inequality, we find the solution set by n(n− 1) n

rewriting the inequality in an equivalent, but simpler, form, using the rules listed in item 1.

= 1 + nx + x 2 + · · ·+ x .2!

2. If a, b ∈ R and n ∈ N, then n

is defined by k=0

a, if a ≥ 0,n−1b +

n(n− 1) n−2b2= a n + na a

3 If a ∈ R, then its modulus or absolute value n n−k bk a(a + b)n = k|a|

|a| = −a, if a < 0. 2! n

The distance on the real line from a to b is |a− b|. + · · ·+ b .

4 Properties of the modulus

If a, b ∈ R, then 3 Inequalities for real numbers )2 a + b1. |a| ≥ 0, with equality if and only if a = 0; (1) ab ≤ , for a, b ∈ R.

22. a| ≤ a ≤ |a|; √−|3. |a|2 = a2;

a− b

(2) 2 ≤ a + b, for a, b ≥ 0. √ √ a2 + b

(3) , for a, b ≥ 0.| a− b| ≤ |a− b|4. ;||ab||

| = |b− a|| |b

< b ⇔ −b < a < b

45. 6.

Inequalities for natural numbers ;| |= a

(1) 2n ≥ 1 + n, for n ≥ 1.|a . 1

(2) 21/n ≤ 1 + , for n ≥ 1. n

(3) 2n ≥ n2, for n ≥ 4. (4) 21/n ≥ 2n/(2n− 1), for n ≥ 1.

5 Bernoulli’s Inequality For n ∈ N, (1 + x)n ≥ 1 + nx, for x ≥ −1.

61

′

′ ′

AA1

6 Strategies for proving inequalities

(1) Give a direct proof. (2) Use the Binomial Theorem. (3) Use mathematical induction. (4) Deduce the result from a known inequality.

4 Least upper bounds

1 A set E ⊆ R is bounded above if there is a real number M , called an upper bound of E, such that

x ≤ M, for all x ∈ E.

If the upper bound M belongs to E, then M is called

6 Least Upper Bound Property of R Let E be a non-empty subset of R. If E is bounded above, then E has a least upper bound. Greatest Lower Bound Property of R Let E be a non-empty subset of R. If E is bounded below, then E has a greatest lower bound.

5 Manipulating real numbers

1 We define the sum and product of two positive real numbers a and b (expressed as decimals) as follows. Form the sums (or products) of truncations of a and b to n decimal places for each n ∈ N, and

the maximum element of E, denoted by max E.

2 A set E ⊆ R is bounded below if there is a real number m, called a lower bound of E, such that

m ≤ x, for all x ∈ E.

If the lower bound m belongs to E, then m is called the minimum element of E, denoted by min E.

3 A set E ⊆ R is bounded if E is bounded above and bounded below; the set E is unbounded if it is not bounded.

4 A real number M is the least upper bound, or supremum, of a set E ⊆ R if 1. M is an upper bound of E; 2. each M ′ < M is not an upper bound of E. In this case, we write M = sup E.

Strategy 4.1 Given a subset E of R, to show that M is the least upper bound, or supremum, of E, check that: 1. x ≤ M , for all x ∈ E; 2. if M ′ < M , then there is some x ∈ E such

that x > M ′ .

5 A real number m is the greatest lower bound, or infimum, of a set E ⊆ R if 1. m is a lower bound of E; 2. each m > m is not a lower bound of E.

In this case, we write m = inf E.

Strategy 4.2 Given a subset E of R, to show that m is the greatest lower bound, or infimum, of E, check that: 1. x ≥ m, for all x ∈ E; 2. if m > m, then there is some x ∈ E such

that x < m .

take the least upper bound of the resulting set offinite decimals.Similar ideas can be used to define the operations ofsubtraction and division.

2 Existence of roots

Theorem 5.1 For each positive real number a and each integer n > 1, there is a unique positive real number b such that

bn = a.

We call this positive number b the nth root of a,√ nand write b = a.

3 If a > 0, m ∈ Z and n ∈ N, then we define √ am/n = ( n a)m .

4 Exponent Laws If a, b > 0 and x, y ∈ Q, then x axb = (ab)x ,

xa ay = a x+y ,x)y = axy(a .

62

AA2

AA2 Sequences 3 When a given sequence has a certain property, provided that we ignore a finite number of terms, we say that the sequence eventually has this property.

1 Introducing sequences 2 Null sequences

1 A sequence is an unending list of real numbers 1 The sequence an is null ifa1, a2, a3, . . . .

The real number an is called the nth term of the sequence, and the sequence is denoted by

an.

for each positive number ε, there is an integer N such that |an| < ε, for all n > N. (∗)

Sequences sometimes begin with a term other than a1; for example, a0 or a3. The sequence diagram of a sequence an is the graph of the sequence in R2, that is, the set of points (n, an) : n = 1, 2, . . ..

2 A sequence an is: constant if

an+1 = an, for n = 1, 2, . . . ; increasing if

an+1 ≥ an, for n = 1, 2, . . . ; strictly increasing if

an+1 > an, for n = 1, 2, . . . ; decreasing if

an+1 ≤ an, for n = 1, 2, . . . ; strictly decreasing if

Strategy 2.1 1. To show that an is null, solve |an| < ε to

find N (depending on ε) for which (∗) holds. 2. To show that an is not null, find one

value of ε for which there is no integer N such that (∗) holds.

an+1 < an, for n = 1, 2, . . . ; 2 The sequence an is null if and only if the sequence |an| is null, and also if and only if the monotonic if an is either increasing or decreasing;sequence (−1)nan is null.

strictly monotonic if an is either strictly increasing or strictly decreasing.

Strategy 1.1 To show that a given sequence an is monotonic, consider the difference an+1 − an. If an+1 − an ≥ 0, for n = 1, 2, . . . , then an is increasing. If an+1 − an ≤ 0, for n = 1, 2, . . . , then an is decreasing.

Strategy 1.2 To show that a given sequence an of positive terms is monotonic, consider the quotient

an+1 .an

If an+1 ≥ 1, for n = 1, 2, . . . , then an is an

increasing.

If an+1 ≤ 1, for n = 1, 2, . . . , then an is an

decreasing.

The null sequence an remains null if we add, delete or alter a finite number of terms.

Power Rule If an is null, where an ≥ 0 for apn = 1, 2, . . . , and p > 0, then n is null.

Combination Rules If an and bn are null, then: Sum Rule an + bn is null; Multiple Rule λan is null, for λ ∈ R; Product Rule anbn is null.

63

)

AA2

Squeeze Rule If bn is null and

|an| ≤ bn, for n = 1, 2, . . .

(that is, an is dominated by bn), thenan is null.

Strategy 2.2 Squeeze Rule To show that an is null. 1. Guess a dominating null sequence bn. 2. Check that |an| ≤ bn for n = 1, 2, . . . , using

the rules for inequalities. 3. If |an| ≤ bn for n = 1, 2, . . . , then an is

null, since bn is null.

3 Basic null sequences The following sequences are null. (a) 1/np, for p > 0.

n(b) c , for |c| < 1. n(c) npc , for p > 0, |c| < 1.

(d) cn/n!, for c ∈ R. (e) np/n!, for p > 0.

3 Convergent sequences

1 The sequence an is convergent with limit l if an − l is a null sequence. We say that anconverges to l, and we write

either lim an = l, n→∞

or an → l as n →∞. Equivalently, the sequence an converges to l if

for each positive number ε, there is an integer N such that |an − l| < ε, for all n > N.

2 If a sequence is convergent, then it has a unique limit. If a given sequence converges to l, then this remains true if we add, delete or alter a finite number of terms.

Combination Rules If lim an = l and n→∞

lim bn = m, then:n→∞

Sum Rule lim (an + bn) = l + m; n→∞

Multiple Rule lim (λan) = λl, for λ ∈ R; n→∞

Product Rule lim (anbn) = lm; n→∞(

Quotient Rule lim an =

l ,

n→∞ bn m provided that m = 0.

Strategy 3.1 To evaluate the limit of a complicated quotient. 1. Identify the dominant term. 2. Divide both numerator and denominator by

the dominant term. 3. Apply the Combination Rules.

When applying Strategy 3.1, n n! dominates c ,

and, for |c| > 1 and p > 0, n c dominates np.

Squeeze Rule If an, bn and cn are sequences such that:

1. bn ≤ an ≤ cn, for n = 1, 2, . . . ,

2. lim bn = lim cn = l, n→∞ n→∞

then lim an = l. n→∞

3 For any positive number a, lim a 1/n = 1.

n→∞

Also, lim n 1/n = 1.

n→∞

Limit Inequality Rule If lim an = l and n→∞

lim bn = m, and also n→∞

an ≤ bn, for n = 1, 2, . . . ,

then l ≤ m.

64

AA2

Corollary If lim an = l and lim an = m, n→∞ n→∞

then l = m.

Theorem 3.1 If lim an = l, then n→∞

lim |an| = |l|. n→∞

4 Divergent sequences

1 A sequence is divergent if it is not convergent. A sequence an is bounded if there is a number K such that |an| ≤ K, for n = 1, 2, . . . .

A sequence is unbounded if it is not bounded.

Theorem 4.1 If an is convergent, then an is bounded.

Corollary If an is unbounded, then anis divergent.

2 The sequence an tends to infinity if for each positive number K, there is an integer N such that

an > K, for all n > N.

In this case, we write

an →∞ as n →∞.

The sequence an tends to minus infinity if −an →∞ as n →∞.

In this case, we write

an → −∞ as n →∞.

If a sequence tends to ∞ or −∞, then it is unbounded, and hence divergent.If a sequence tends to ∞ or −∞, then this remainstrue if we add, delete or alter a finite number ofterms.

Reciprocal Rule If the sequence ansatisfies both the conditions 1. an is eventually positive, 2. 1/an is a null sequence, then an →∞.

Combination Rules If an tends to infinity and bn tends to infinity, then: Sum Rule an + bn tends to infinity; Multiple Rule λan tends to infinity,

for λ ∈ R+; Product Rule anbn tends to infinity.

Squeeze Rule If bn tends to infinity and

an ≥ bn, for n = 1, 2, . . . ,

then an tends to infinity.

3 The sequence ank is a subsequence of the sequence an if nk is a strictly increasing sequence of positive integers; that is, if

n1 < n2 < n3 < · · · . In particular, a2k is the even subsequence and a2k−1 is the odd subsequence. Every sequence is a subsequence of itself.

Theorem 4.2 For any subsequence ank of an: (a) if an → l as n →∞,

then ank → l as k →∞;(b) if an →∞ as n →∞,

then ank →∞ as k →∞.

Corollary 1. First Subsequence Rule The sequence

an is divergent if an has two convergent subsequences with different limits.

2. Second Subsequence Rule The sequence an is divergent if an has a subsequence which tends to infinity or a subsequence which tends to minus infinity.

Strategy 4.1 To prove that the sequence an is divergent: either 1. show that an has two convergent

subsequences with different limits;or 2. show that an has a subsequence which

tends to infinity or a subsequence which tends to minus infinity.

65

( )

∑

∑

AA3

Theorem 4.3 Let an consist of two subsequences amk and ank , which both tend to the same limit l. Then

lim an = l. n→∞

5 Monotone Convergence Theorem

1 Monotonic sequences

Theorem 5.1Monotone Convergence TheoremIf the sequence an is:either increasing and bounded above,or decreasing and bounded below,then an is convergent.

Theorem 5.2 Monotonic Sequence Theorem If the sequence an is monotonic, then either an is convergent or an → ±∞.

AA3 Series

1 Introducing series

1 Let an be a sequence. Then the expression

a1 + a2 + a3 + · · · is an infinite series, or simply a series. We call an

the nth term of the series.The nth partial sum of this series is

sn = a1 + a2 + · · ·+ an.

2 The series a1 + a2 + a3 + · · ·

is convergent with sum s if its sequence sn of partial sums converges to s. In this case, the series converges to s and we write

a1 + a2 + a3 + · · · = s.

The series diverges, or is divergent, if the sequencesn diverges.

3 Sigma notation We write∞∑

an = a1 + a2 + a3 + · · · . n=1

When using sigma notation to represent the nth partial sum of such a series, we write

n

sn = a1 + a2 + · · ·+ an = ak. k=1

If we need to begin a series with a term other than a1, we write, for example,

∞∑ an = a0 + a1 + a2 + · · · ;

n=0

for such a series, n

sn = a0 + a1 + · · ·+ an = ak. k=0

4 Geometric series

The (infinite) geometric series with first term a and common ratio r is

∞∑ n

2 We define the number π as follows: ar = a + ar + ar 2 + · · · . n=0

π = lim an = 3.141 592 653 . . . , n→∞

where an is the area of the regular polygon with Geometric series 3 × 2n sides inscribed in a disc of radius 1. ∞∑

nWe define the number e as follows: (a) If |r| < 1, then ar is convergent, with ( )n1 e = lim 1 + = 2.718 281 828 . . . . sum

a .

n=0

n→∞ n 1 − rIf x is irrational and x ≥ 0, then lim (1 + x/n)n

n→∞ (b) If |r| ≥ 1 and a = 0, then exists and is used to define ex:

divergent.

∞∑ n isar

n=0 nx

e = lim 1 + . n→∞ n

x

66

AA3

5 A telescoping series is one for which it is easy 2 Series with non-negative terms to find an expression for sn because most terms in the partial sums cancel out. 1 Tests for convergence

Combination Rules Suppose that Comparison Test ∞∑∞∑ an = s and bn = t. Then (a) If

n=1 n=1 0 ≤ an ≤ bn, for n = 1, 2, . . . , ∞∑

Sum Rule (an + bn) = s + t; and ∞∑∞∑

bn is convergent, then an is n=1

n=1 n=1 ∞∑ Multiple Rule λan

n=1

convergent. = λs, for λ ∈ R.

(b) If0 ≤ bn ≤ an, for n = 1, 2, . . . ,

∞∑∞∑ and bn is divergent, then an is

n=1 n=1∞∑

Theorem 1.1 If an is a convergent divergent. n=1

series, then its sequence of terms an is a null sequence. The Comparison Test tells us that any series of the

form ∞∑

a0 + Corollary Non-null Test n=1

an

10n ,

∞∑ where a0 is a non-negative integer and an, n = 1, 2, . . . , are digits, must be convergent.If an is not a null sequence, then an is

n=1 Moreover, the partial sums of this series are divergent. sn = a0.a1a2 . . . an, so its sum is a0.a1a2 . . . . Thus this series provides an alternative interpretation of the infinite decimal a0.a1a2 . . . .

Strategy 1.1 ∞∑

To show that an is divergent using the Limit Comparison Test Suppose that n=1

Non-null Test. ∞∑∞∑

an and bn have positive terms and that either n=1 n=1

1. show that |an| has a convergent an → L as n →∞, subsequence with non-zero limit, bn

or where L = 0. ∞∑∞∑

bn is convergent, then an is2. show that |an| has a subsequence which

tends to infinity. (a) If n=1 n=1

convergent. ∞∑∞∑

bn is divergent, then an isWarning You can never use the Non-null Test to (b) If prove that a series is convergent.

n=1 n=1 divergent.

67

∣ ∣ ∣ ∣ ∣∣ ∣ ∣ ∣ ∣

∑

∑

∑

∑ ∑

∑

∑

∑

AA3

∞∑ Ratio Test Suppose that an has positive

Alternating Test Let = (−1)n+1bn, n = 1, 2, . . . , an

n=1

terms and that an+1 → l as n →∞. where bn is a decreasing null sequence with an positive terms. Then

∞∑ ∞∑(a) If 0 ≤ l < 1, then an is convergent. an = b1 − b2 + b3 − b4 + · · ·

n=1 n=1

is convergent. ∞∑ (b) If l > 1, then an is divergent.

n=1

If l = 1, then the Ratio Test is inconclusive. ∞∑

Strategy 3.1 To prove that an is n=12 Basic series The following series are

convergent: convergent using the Alternating Test, check that∞∑ 1

(a) , for p ≥ 2; an = (−1)n+1bn, n = 1, 2, . . . , np

n=1 where

1. bn ≥ 0, for n = 1, 2, . . . ; ∞∑

(b) c n, for 0 ≤ c < 1; n=1 2. ∞∑ bn is a null sequence;

3. is decreasing. (c) npc n, for p > 0, 0 ≤ c < 1; bn n=1 ∞∑ nc

(d) , for c ≥ 0.n! Strategy 3.2 To test the series a forn=1 n

The following series is divergent: convergence or divergence. ∞∑ 1. If you think that the sequence of terms an1

, for 0 < p ≤ 1. is non-null, then try the Non-null Test.(e) np

n=1 2. If an has non-negative terms, then try

one of these tests. 3 Series with positive and negative (a) Basic series Is an a basic series, or terms a combination of these?

(b) Comparison Test b is convergent, or a

Is an ≤ bn, where ∞∑ The series an is absolutely convergent if ≥ bn ≥ 0,n n

where bn is divergent? n=1∞∑

is convergent. (c) Limit Comparison Test Does abehave like b

|an| n

n (that is, an/bn → L = 0), n=1 where bn is a basic series?

(d) Ratio Test Does an+1/an → l = 1? Absolute Convergence Test

3. If an has positive and negative terms, ∞∑∞∑ If an is absolutely convergent, then an is then try one of these tests.

(a) Absolute Convergence Test |an| convergent? (Use step 2.)

(b) Alternating Test Is an = (−1)n+1bn, where bn is non-negative, null and decreasing?

Isn=1 n=1

convergent.

Triangle Inequality (infinite form)∞∑

If an is absolutely convergent, then n=1

∞∑ ∞∑ an ≤ |an|.

n=1 n=1

68

The following suggestions may also be helpful. If an is positive and includes n! or cn, then considerthe Ratio Test.If an is positive and has dominant term np, then consider the (Limit) Comparison Test.If an includes a sine or cosine term, then use the factthat this term is bounded and consider theComparison Test and the Absolute Convergence Test.

4 Exponential function

Theorem 4.1 If x > 0, then ∞∑ xn ( x )n

= lim 1 + = e x . n! n→∞ n

n=0

For x ≥ 0, we define ∞( )n ∑ xnxx e = lim 1 + = .

n→∞ n n! n=0

For x < 0, we define x −x)−1 e = (e .

The exponential function x −→ ex is often called exp. Thus

exp : R −→ R

x −→ e x .

Theorem 4.2 The number e is irrational.

Theorem 4.3 For any real numbers x and y, x+y xey .we have e = e

AA4

AA4 Continuity

1 Operations on functions

1 Convention When a function f is specified just by its rule, it is to be understood that the domain of f is the set of all real numbers for which the rule is applicable and the codomain of f is R. A function f is defined on a set I (usually an interval) if the domain of f contains the set I. If a, b ∈ R, then

(a, b), (a,∞ ) and (−∞ , b) are open intervals; [a, b], [a,∞ ) and (−∞ , b] are closed intervals; [a, b) and (a, b] are half-open intervals; R = (−∞ ,∞ ) is both open and closed.

2 If f : A −→ R and g : B −→ R, then

the sum f + g is the function with domain A ∩ B and rule

(f + g)(x) = f(x) + g(x); the multiple λf is the function with domain A and rule

(λf)(x) = λf(x), for λ ∈ R; the product fg is the function with domain A ∩ B and rule

(fg)(x) = f(x)g(x); the quotient f/g is the function with domain A ∩ B − x : g(x) = 0 and rule

(f/g)(x) = f(x)/g(x); the composite g f is the function with domain x ∈ A : f(x) ∈ B and rule

(g f)(x) = g(f(x)).

3 Let f : A −→ R be a one-one function. Then the inverse function f−1 has domain f(A) and rule

f−1(y) = x, where y = f(x).

4 A function f defined on an interval I is increasing on I if

x1 < x2 ⇒ f(x1) ≤ f(x2), for x1, x2 ∈ I; strictly increasing on I if

x1 < x2 ⇒ f(x1) < f(x2), for x1, x2 ∈ I; decreasing on I if

x1 < x2 ⇒ f(x1) ≥ f(x2), for x1, x2 ∈ I; strictly decreasing on I if

x1 < x2 ⇒ f(x1) > f(x2), for x1, x2 ∈ I; monotonic on I if f is either increasing on I ordecreasing on I;strictly monotonic on I if f is either strictlyincreasing on I or strictly decreasing on I.A strictly monotonic function f is one-one.

69

AA4

2 Continuous functions

1 A function f : A −→ R is continuous at a (a ∈ A) if

for each sequence xn in A such that xn → a, f(xn) → f(a).

We can write the above condition more concisely, as follows:

xn → a ⇒ f(xn) → f(a), where xn lies in A.

We say that f is continuous (on A) if f is continuous at each a ∈ A.

Strategy 2.1 1. To show that f is continuous at a (a ∈ A),

prove thatfor each sequence xn in Asuch that xn → a,f(xn) → f(a).

2. To show that f is discontinuous at a

Squeeze Rule Let f , g and h be defined on an open interval I and let a ∈ I. If

1. g(x) ≤ f(x) ≤ h(x), for x ∈ I, 2. g(a) = f(a) = h(a), 3. g, h are continuous at a, then f is also continuous at a.

Glue Rule Let f be defined on an open interval I and let a ∈ I. If there are functions g and h such that 1. f(x) = g(x), for x ∈ I, x < a,

f(x) = h(x), for x ∈ I, x > a, 2. g(a) = f(a) = h(a), 3. g and h are continuous at a, then f is also continuous at a.

(a ∈ A), find one sequence xn in A such that

xn → a but f(xn) → f(a).

2 Rules for continuity

Combination Rules If f and g arecontinuous at a, then so are:

Sum Rule f + g;Multiple Rule λf, for λ ∈ R;Product Rule fg;Quotient Rule f/g, provided that g(a) = 0.

We say that continuity at a point is a local property ; it depends only on the values taken by the function near the point. If a function f is the restriction of another function g, and g is continuous, then f is also continuous.

3 Trigonometric and exponential functions

Sine Inequality Composition Rule If f is continuous at a sin x ≤ x, for 0 ≤ x ≤ 1

2π. and g is continuous at f(a), then g f iscontinuous at a.

Corollary | sin x| ≤ | x| , for x ∈ R.

70

1

Exponential Inequalities

(a) ex ≥ 1 + x, for x ≥ 0;

x ≤(b) e , for 0 ≤ x < 1.1 − x

1Corollary 1 + x ≤ ex ≤ , for |x| < 1.

1 − x

Basic continuous functions The following functions are continuous: 1. polynomials and rational functions; 2. f(x) = |x|;√3. f(x) = x; 4. trigonometric functions (sine, cosine and

tangent); 5. the exponential functions.

3 Properties of continuous functions

1 We say that a function f is continuous on an interval I if f is continuous at each point of I.

Theorem 3.1Intermediate Value TheoremLet f be a function continuous on [a, b] and let k be any number lying between f(a) and f(b). Then there exists a number c in (a, b) such that

f(c) = k.

AA4

Theorem 3.2 Antipodal Points Theorem If g : [0, 2π] −→ R is a continuous function and g(0) = g(2π), then there exists a number c in [0, π] such that

g(c) = g(c + π).

3 Locating zeros of polynomial functions

By the Fundamental Theorem of Algebra, a polynomial of degree n has at most n zeros.

Theorem 3.3 Let p(x) = x n + an−1x n−1 + · · ·+ a1x + a0,

where a0, a1, . . . , an−1 ∈ R. Then all the zeros of p (if there are any) lie in the open interval (−M,M), where

M = 1 + max|an−1|, . . . , |a1|, |a0|.

4 Let f be a function with domain A. Then

f has maximum value f(c) in A if c ∈ A and

f(x) ≤ f(c), for x ∈ A; f has minimum value f(c) in A if c ∈ A and

f(c) ≤ f(x), for x ∈ A; f is bounded on A if, for some M ∈ R, |f(x)| ≤ M, for x ∈ A.

An extreme value is a maximum or minimum value.

Theorem 3.4 Extreme Value Theorem Let f be a function continuous on [a, b]. Then there exist numbers c and d in [a, b] such that

f(c) ≤ f(x) ≤ f(d), for x ∈ [a, b].

2 If f is a function and c is a number such that f(c) = 0, then c is a zero of f , and f vanishes at c.

Bisection Method To locate a zero of a continuous function f approximately, find numbers a and b such that f(a) and f(b) have opposite signs; then f has a zero in (a, b). Now use repeated bisection of (a, b), applying this argument to smaller and smaller intervals.

Theorem 3.5 Boundedness Theorem Let f be a function continuous on [a, b]. Then there exists a number M such that |f(x)| ≤ M, for x ∈ [a, b].

71

AA4

4 Inverse functions

1 Finding inverse functions

Inverse Function Rule Let f : I −→ J , where I is an interval and J is the image f(I), be a function such that 1. f is strictly increasing on I; 2. f is continuous on I. Then J is an interval and f has an inversefunction f−1 : J −→ I such that1′ . f−1 is strictly increasing on J ;2′ . f−1 is continuous on J .

The graph of f−1 is obtained by reflecting the graph of f in the line y = x. There is another version of the Inverse Function Rule, with ‘strictly increasing’ replaced by ‘strictly decreasing’.

Strategy 4.1 To prove that f : I −→ J , where I is an interval with endpoints a and b, has a continuous inverse f−1 : J −→ I. 1. Show that f is strictly increasing on I. 2. Show that f is continuous on I. 3. Determine the endpoint c of J

corresponding to the endpoint a of I asfollows:

if a ∈ I, then f(a) = c and c ∈ J , ∈ I, then f(an) → c and c /if a / ∈ J ,

where an is a monotonic sequence in I such that an → a.

Similarly, determine the endpoint d of J , corresponding to the endpoint b of I.

There is a corresponding version of this strategy if f is strictly decreasing.

2 For any positive integer n ≥ 2, the function nf(x) = x (x ∈ [0,∞ ))

has a strictly increasing continuous inverse function √ f−1(x) = n x, with domain [0,∞ ) and image [0,∞ ), called the nth root function.

3 Inverse trigonometric functions

The function sin−1 The function

f(x) = sin x (x ∈ [− 1 22 π, 1 π]) has a strictly increasing continuous inverse function, sin−1, with domain [− 1, 1] and image [− 1

2 π, 1 π].2

The function cos−1 The function

f(x) = cos x (x ∈ [0, π]) has a strictly decreasing continuous inverse function, cos−1, with domain [− 1, 1] and image [0, π]. The function tan−1 The function

f(x) = tan x (x ∈ (− 1 22 π, 1 π)) has a strictly increasing continuous inverse function, tan−1, with domain R and image (− 1 22 π, 1 π).

4 The function loge The function xf(x) = e (x ∈ R)

has a strictly increasing continuous inverse function, with domain (0,∞ ) and image R, called loge or ln. For all x, y > 0,

loge x + loge y = loge xy.

5 Inverse hyperbolic functions

The function sinh−1 The function 1f(x) = sinh x = 2 (e x − e −x) (x ∈ R)

has a strictly increasing continuous inverse function, sinh−1, with domain R and image R. The function cosh−1 The function

1f(x) = cosh x = 2 (e x + e −x) (x ∈ [0,∞ )) has a strictly increasing continuous inverse function, cosh−1, with domain [1,∞ ) and image [0,∞ ).

The function tanh−1 The function sinh x

f(x) = tanh x = (x ∈ R)cosh x

has a strictly increasing continuous inverse function, tanh−1, with domain (− 1, 1) and image R.

6 Defining exponential functions

If a > 0, then

a x = e x loge a (x ∈ R).

Theorem 4.1 (a) If a > 0, then the function

x −→ a x = e x loge a (x ∈ R) is continuous.

(b) If a, b > 0 and x, y ∈ R, then x a xb = (ab)x ,

a x ay = a x+y ,x)y = axy(a .

72

Group Theory Block B

GTB1 Conjugacy

2 Conjugate elements

1 Let x and y be elements of a group G; then y is a conjugate of x in G if there exists an element g ∈ G such that

−1 y = gxg .

We then also say that g conjugates x to y, that g is a conjugating element and that y is the conjugateof x by g.(Here we have omitted the symbol for the groupoperation: we have written gxg−1 rather thang x g−1 . For convenience, we often omit thesymbol in this block.)

Theorem 2.2 In any group G, the relation is conjugate to is an equivalence relation on the set of elements of G.

The symmetric property of conjugacy means that instead of saying that y is a conjugate of x, we can simply say that x and y are conjugate elements in a group, meaning that each is a conjugate of the other. Also, if g conjugates x to y, then g−1 conjugates y to x.

2 The equivalence classes of the equivalence relation is conjugate to on a group G are called the conjugacy classes of G. The conjugacy class of an element x ∈ G can be written in set notation as y ∈ G : y = gxg −1 , for some g ∈ G

or as gxg −1 : g ∈ G.

Thus if two elements are conjugate, then they are in the same conjugacy class in G; if they are not conjugate, then they are in different conjugacy classes. Each group element belongs to one and only one conjugacy class.

Theorem 2.3 Let G be a group with identity element e. Then e is a conjugacy class; that is, e is conjugate to itself alone.

GTB1

Theorem 2.4 In an Abelian group, each conjugacy class contains a single element.

n3 If g conjugates x to y, then g conjugates xto yn, for each positive integer n.

Theorem 2.5 Let x and y be conjugate elements in a group G; then x and y have the same order.

The converse of Theorem 2.5 is not true: elements of the same order are not necessarily conjugate.

4 Informally, we say that two symmetries x and y of a geometric figure are of the same geometric type when there is a symmetry of the figure that transforms a diagram illustrating x into a diagram illustrating y (when we ignore any labels).

5 There are three conjugacy classes in S(): e, a, b, r, s, t.

There are five conjugacy classes in S(): e, b, a, c, r, t, s, u.

These partitions relate elements of the same geometric type.

6 Conjugacy of permutations is covered on pages 33–34.

3 Normal subgroups and conjugacy

1 Notation Let H be a subgroup of a group G and let g be any element of G. Then

gHg −1 = ghg −1 : h ∈ H .

Theorem 3.2 Let H be a subgroup of a group G and let g be any element of G. Then the subset gHg−1 is a subgroup of G.

Two subgroups H and H ′ of a group G are conjugate subgroups in G if there exists an element g ∈ G such that

H ′ = gHg −1 .

A subgroup H of a group G is self-conjugate if gHg −1 = H for all g ∈ G.

73

GTB1

2 Characterisations of normality

Theorem 3.5 Let N be a subgroup of a group G. The following four properties are equivalent. A: gN = Ng, for each g ∈ G. B: gng−1 ∈ N, for each g ∈ G and each n ∈ N. C: gNg−1 = N, for each g ∈ G. D: N is a union of conjugacy classes of G.

Since Property A is the condition in the definition of a normal subgroup, Theorem 3.5 tells us that a subgroup N of a group G is a normal subgroup of G if and only if it satisfies any one of the four properties A, B, C and D. Any of the four properties can be used to prove that a subgroup is normal, or to show that it is not normal. Property A is useful when information concerning the left and right partitions into cosets is known. Property B is useful in many general situations. Property C is helpful when information about conjugate subgroups is known. Property D is particularly useful when the conjugacy classes are known.

3 Property B gives the following strategies.

Strategy 3.2 To prove that a subgroup N is a normal subgroup of a group G. 1. Take a general element g ∈ G and a general

element n ∈ N . 2. Show that the conjugate gng−1 belongs

to N .

Strategy 3.3 To prove that a subgroup H is not a normal subgroup of a group G. Find one element g ∈ G and one element h ∈ H such that the conjugate ghg−1 does not belong to H.

4 The equivalence of Properties A and C means that normal subgroups and self-conjugate subgroups are the same objects.

5 Property D gives the following strategy.

Strategy 3.4 To find all the normal subgroups of a finite group G, when the partition of G into conjugacy classes is known. 1. Look at all the possible unions of conjugacy

classes that include the class e. 2. Consider only those unions for which the

total number of elements divides |G|, the order of the group G.

3. Determine whether each union of conjugacy classes is a subgroup of G: any union that is a subgroup is a normal subgroup of G.

6 The normal subgroups of S() are

S(), e, a, b, e. The normal subgroups of S() are

S(), e, a, b, c, e, b, r, t, e, b, s, u, e, b, e.

The normal subgroups of S4 are

S4, A4, K4, e.

4 Conjugacy in symmetry groups

1 If the symmetries in a symmetry group G are represented as permutations, then elements with different cycle structures lie in different conjugacy classes, and elements with the same cycle structure may, or may not, lie in the same conjugacy class.

2 Let f be a symmetry of a figure F . Then the fixed point set of f is x ∈ F : f(x) = x,

and is denoted by Fix(f); that is, the fixed point set of f is the set of points of the figure which are fixed by f .

Theorem 4.1 Fixed Point Theorem Let g and k be elements of a symmetry group G. If L is the fixed point set of g, then k(L) is the fixed point set of the conjugate element kgk−1 .

The Fixed Point Theorem applies also to any group whose elements are functions of some kind; in particular, it holds for permutation groups, where the fixed points become fixed symbols.

74

( )

( )

( )

( )

( )

GTB2

3 One approach to finding the conjugacy class of a given element s of a symmetry group is as follows. 1. Find each symmetry t for which there is a

symmetry k that maps Fix(s) to Fix(t). Each such symmetry t may be a conjugate of s. Any other symmetry is not a conjugate of s.

2. To test whether each such symmetry t is a conjugate of s, try checking whether t = ksk−1 , where k is a symmetry that maps Fix(s) to Fix(t). (If t = ksk−1, then t may or may not be a conjugate of s.)

Theorem 4.2 A direct symmetry cannot be conjugate to an indirect symmetry in a symmetry group.

5 Matrix groups

GTB2 Homomorphisms

1 Isomorphisms and homomorphisms

1 A homomorphism is a function φ : (G, ) −→ (H, ∗ ), where (G, ) and (H, ∗ ) are groups, which has the property

φ(g1 g2) = φ(g1) ∗ φ(g2), for all g1, g2 ∈ G.

This property is called the homomorphism property. A function satisfying it is said to preservecomposites.If the homomorphism property is satisfied, then itextends to products of three or more elements: forexample, if φ : (G, ) −→ (H, ∗ ) is a homomorphismand g1, g2, g3 ∈ G, then

φ(g1 g2 g3) = φ(g1) ∗ φ(g2) ∗ φ(g3). A homomorphism need not be either one-one or onto. A homomorphism that is one-one and onto is an isomorphism.

1 The set M of all invertible 2 × 2 matrices with real entries is a group under matrix multiplication. The identity element of M is

1 0I = 0 1

.

a bThe inverse of A = is c d

1 d − bA−1 = , where ad − bc = 0.

ad − bc − c a

2 Among the subgroups of M are the following. Upper-triangular matrices:

Strategy 1.1 To determine whether a given function φ : (G, ) −→ (H, ∗ ) is a homomorphism. guess behaviour, . . . check definition. To show that φ is a homomorphism, show, by a general argument, that

φ(g1 g2) = φ(g1) ∗ φ(g2), for all g1, g2 ∈ G.

To show that φ is not a homomorphism, find any two elements g1, g2 ∈ G such that

φ(g1 g2) = φ(g1) ∗ φ(g2).

a b = 0 . 2 If V and W are vector spaces, then any linear 0 d transformation from V to W is a homomorphism

Lower-triangular matrices: from the group (V, +) to the group (W, +).

U = : ad

a 0 = 0 . 3 If F is a figure with symmetry group S(F ), then L = : ad c d the function

Matrices with determinant 1: ( ) φ : (S(F ), ) −→ (R ∗ , × )a b

V = : ad − bc = 1 . 1, if f is a direct symmetry,c d f −→ − 1, if f is an indirect symmetry,

3 Further standard associative operations is a homomorphism. The following operations are associative and may be 4 If (G, ) and (H, ∗ ) are groups and eH is the quoted as such: identity element of H, then the function

addition of matrices; φ : (G, ) −→ (H, ∗ ) multiplication of matrices. g −→ eH

is a homomorphism, called the trivial homomorphism.

5 An automorphism is an isomorphism from a group to itself. The set of all automorphisms of a group forms a group itself, under composition of functions. The identity of this group is the identity function, which maps each element to itself.

75

GTB2

2 Properties of homomorphisms

Property 2.1 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then

φ(eG) = eH ,

where eG is the identity in (G, ) and eH is the identity in (H, ∗ ).

Property 2.2 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then, for all g ∈ G,

φ(g −1) = (φ(g))−1 .

Property 2.3 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then, for all g ∈ G and all n ∈ Z,

φ(g n) = (φ(g))n .

Theorem 2.2 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism and let g be any element of finite order in G. Then the order of φ(g) divides the order of g.

Property 2.4 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. If x and y are conjugate in G, then φ(x) and φ(y) are conjugate in H.

3 Kernels and images

1 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then the image of φ is

Im(φ) = h ∈ H : h = φ(g) for some g ∈ G ; it is the set of elements of the codomain H which occur as images of elements in the domain G.

2 Properties of images

Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. 1. For each h ∈ H, there may be more than one

g ∈ G such that φ(g) = h, since φ may be many-one.

2. Im(φ) = H ⇔ φ is onto. 3. eH ∈ Im(φ).

Theorem 3.1 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then (Im(φ), ∗ ) is a subgroup of (H, ∗ ).

Theorem 3.2 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. (a) If (G, ) is Abelian, then (Im(φ), ∗ ) is

Abelian.(b) If (G, ) is cyclic, then (Im(φ), ∗ ) is cyclic.

3 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then the kernel of φ is

Ker(φ) = g ∈ G : φ(g) = eH ; it is the set of elements of the domain G which φ maps to eH , the identity element in the codomain H.

Strategy 3.1 To find the kernel of a homomorphism. 1. Identify eH , the identity in the codomain H. 2. Find all the elements g in the domain G

which are mapped to eH .

4 Properties of kernels

Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. 1. There may be more than one g ∈ G such that

φ(g) = eH , since φ may be many-one. 2. eG ∈ Ker(φ).

Theorem 3.3 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then Ker(φ) is a normal subgroup of (G, ).

76

GTB2

Theorem 3.4 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then

φ is one-one ⇔ Ker(φ) = eG .

Strategy 3.2 To show that a homomorphism φ : (G, ) −→ (H, ∗ ) is one-one, show that Ker(φ) = eG.

Corollary to Theorem 3.4 Ker(φ) = eG⇔ (G, ) ∼= (Im(φ), ∗ ),

Ker(φ) = eG and φ is onto⇔ (G, ) ∼= (H, ∗ ).

4 The Isomorphism Theorem

1 Because the kernel of a homomorphism is a normal subgroup of its domain, the left cosets of the kernel in the domain are the same as its right cosets, and we refer to them simply as cosets.

Theorem 4.1 Let φ : (G, ) −→ (H, ∗ ) be a homomorphism, and let x and y be any elements of G. Then

x and y have the same image under φ

if and only if x and y lie in the same coset of Ker(φ) in G.

Theorem 4.2 Correspondence Theorem Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then there is a one-one correspondence between the cosets of Ker(φ) in G and the elements of the image set Im(φ), given by

x Ker(φ) ←→ φ(x).

Theorem 4.3 Let (G, ) be a finite group and let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then

| Ker(φ)| × | Im(φ)| = | G| .

2 If (G, ) and (H, ∗ ) are finite groups, then the following numerical relationships hold for any homomorphism φ : (G, ) −→ (H, ∗ ): | Ker(φ)| divides | G| (by Lagrange’s Theorem), | Im(φ)| divides | H| (by Lagrange’s Theorem), | Im(φ)| divides | G| (by Theorem 4.3).

In particular, the order of the image group of a homomorphism is a common factor of the orders of the domain and codomain groups.

Theorem 4.6 Isomorphism Theorem Let φ : (G, ) −→ (H, ∗ ) be a homomorphism. Then the function

f : G/Ker(φ) −→ Im(φ)x Ker(φ) −→ φ(x)

is an isomorphism, so

G/Ker(φ) ∼= Im(φ).

The function f is illustrated in the diagram above.

3 Specifying a familiar group that is isomorphic to a given group G is known as identifying G up to isomorphism.

Theorem 4.7 Let (G, ) be a group. (a) For any homomorphism φ : G −→ H,

Ker(φ) is a normal subgroup of G.(b) For every normal subgroup N of G, the

function

φ : (G, ) −→ (G/N, . )x −→ xN

is a homomorphism with kernel N .

Theorem 4.7 tells us that kernels of homomorphisms and normal subgroups are essentially the same objects.

77

GTB3

GTB3 Group actions

1 What is a group action?

1 Often a group G ‘acts on’ a set X in some way: that is, given any elements g ∈ G and x ∈ X, g sends x to some element of X, which we denote by g ∧ x. To restrict ourselves to actions of interest, we say that G ‘acts on’ X, and that ∧ is a group action, only if certain conditions hold, as set out in the following definition.

2 A group (G, ) acts on a set X if the following three axioms hold. GA1 closure For each g ∈ G and each

x ∈ X, there is a unique element

g ∧ x ∈ X. GA2 identity For each x ∈ X,

e ∧ x = x, where e is the identity element of G.

GA3 composition For all g1, g2 ∈ G and all x ∈ X,

g1 ∧ (g2 ∧ x) = (g1 g2) ∧ x.

We say that ∧ is a group action of G on X. If g ∧ x = y, then we say that g acts on x to give y.

Axiom GA1 states that each element g of the group G acts on each element x of the set X, and the result is always an element of the set X; the action does not take us out of the set X.

Axiom GA2 states that the identity element of the group fixes each element of the set X.

Axiom GA3 states that acting on x successively by two group elements g2 and g1 has the same effect as acting by their composite g1 g2.

Strategy 1.1 To determine whether ∧ is a group action. guess behaviour. . . check definition. To show that ∧ is a group action, show that each of the axioms GA1, GA2 and GA3 holds. To show that ∧ is not a group action, show that any one of the axioms GA1, GA2 or GA3 fails.

3 By the ‘natural’ action of a group G on a set X, we mean the action of G that sends elements of X to other elements of X in the most obvious way.

4 For a group G and a set X that do not contain many elements, we can specify a group action ∧ by recording in an action table (also called a group action table) the effect that it has for each g ∈ G and each x ∈ X. For example, an action table for the natural action of the group S3 on the set 1, 2, 3 is as follows.

78

( )

Theorem 1.1 Let G be a group whose elements are functions from a set X to itself, with the binary operation of composition of functions. Let ∧ be defined by

g ∧ x = g(x), for g ∈ G and x ∈ X.

Then ∧ is a group action of G on X.

In particular, whenever we let a permutation group or a symmetry group act in the natural way as a group of functions on an underlying set, we have a group action.

5 Infinite actions Our definition of group action still applies when one, or both, of the group G and the set X is infinite. In such a case we cannot write out all the action table, but we may usefully be able to present part of it.

Theorem 1.2 Let M be the group of all invertible 2 × 2 matrices, under matrix multiplication. Then

a b ∧ (x, y) = (ax + by, cx + dy)c d

defines a group action of M on R2 .

The group action in Theorem 1.2 is the natural action of M on R2 given by matrix multiplication.

2 Orbits and stabilisers

1 Let a group G act on a set X. Then for each x ∈ X, the orbit of x is the set

Orb(x) = g ∧ x : g ∈ G. Orb(x) is a subset of the set X; it is the set of all elements of X which ‘can be reached from x using the action ∧’.

GTB3

2 Properties of orbits

Let a group G act on a set X. 1. For each x ∈ X, we have x ∈ Orb(x). 2. If y ∈ Orb(x), then x ∈ Orb(y). 3. If y ∈ Orb(x) and z ∈ Orb(y), then z ∈ Orb(x).

That is, the relation on X defined by

x ∼ y if y ∈ Orb(x) is an equivalence relation. The equivalence classes are the orbits.

Theorem 2.1 Let a group G act on a set X. Then the orbits form a partition of X.

In particular, if x, y ∈ X then either Orb(x) and Orb(y) are the same set or Orb(x) and Orb(y) are disjoint.

Strategy 2.1 To find all the orbits in X. 1. Choose any x ∈ X, and find Orb(x). 2. Choose any element of X not yet assigned

to an orbit, and find its orbit. 3. Repeat step 2 until X is partitioned.

It is often useful to look first at the orbits of particular elements, and then try to spot a general pattern for the orbits.

3 Let a group G act on a set X. Then, for each x ∈ X, the stabiliser of x is the set

Stab(x) = g ∈ G : g ∧ x = x. Stab(x) is a subset of the group G; it is the set of all elements of G that fix or stabilise x.

Theorem 2.2 Let a group G act on a set X. Then, for each x ∈ X, the set Stab(x) is a subgroup of G.

79

∑

GTB3

3 The Orbit–Stabiliser Theorem

1 For actions of finite groups we have the following theorem.

Theorem 3.1 Orbit–Stabiliser Theorem Let a finite group G act on a set X. Then, for each x ∈ X, |Orb(x)| × |Stab(x)| = |G|.

Corollary Let a finite group G act on aset X. Then, for each x ∈ X, the number of elements in Orb(x) divides the order of G.

Theorem 3.2 Let ∧ be a group action of G on X, let x ∈ X, and let g and h be any elements of G. Then

g ∧ x = h ∧ x

if and only if g and h lie in the same left coset of Stab(x).

Theorem 3.2 tells us that if G is a group acting on a set X, and x ∈ X, then the sets of group elements that send x to a common element of X are precisely the left cosets of Stab(x).

Corollary Let ∧ be a group action of G on X and let x ∈ X. Then there is a one-one correspondence between the left cosets of Stab(x) in G and the elements of Orb(x), given by

g Stab(x) ←→ g ∧ x.

In items 2 and 3 below we suppress mention of the binary operation of a group G, writing, for example, hg instead of h g.

2 Let H be a subgroup of a group G. Then

h ∧ g = hg, for h ∈ H and g ∈ G,

defines a group action of H on G. The orbits of this group action are precisely the right cosets of H in G. Lagrange’s Theorem (that the order of a subgroup of a finite group G divides the order of G) is a special case of the corollary to the Orbit–Stabiliser Theorem.

3 Let G be a group. Then

g ∧ x = gxg −1 , for g, x ∈ G,

defines a group action of the group G on itself. For any x ∈ G, the set Orb(x) is gxg−1 : g ∈ G , which is the conjugacy class of x. Thus the orbits of this group action are the conjugacy classes of the group G. The following theorem is a special case of the corollary to the Orbit–Stabiliser Theorem.

Theorem 3.3 In any finite group G, the number of elements in any conjugacy class divides the order of G.

4 Let φ : (G, ) −→ (H, ∗) be a homomorphism. Then

g ∧ h = φ(g) ∗ h, for g ∈ G and h ∈ H,

defines a group action of the group G on the set H. In this case, for a finite group G, the statement of the Orbit–Stabiliser Theorem, with x = eH , becomes |Im(φ)| × |Ker(φ)| = |G|.

Thus this consequence of the Correspondence Theorem for homomorphisms is a special case of the Orbit–Stabiliser Theorem.

4 The Counting Theorem

1 The following rule is used to count colourings without taking symmetries into account.

Multiplication Rule If object 1 can be coloured with n1 colours, object 2 can be coloured with n2 colours, . . . , object k can be coloured with nk colours, then the number of ways of colouring all k objects is n1n2 . . . nk.

2 Let a group G act on a set X. For g ∈ G, the fixed set of g is

Fix(g) = x ∈ X : g ∧ x = x. That is, Fix(g) is the set of elements of X that are fixed by g.

Theorem 4.1 Counting Theorem Let ∧ be a group action of a finite group G on a finite set X. Then the number t of orbits of the action is given by the formula

1 t = |G| |Fix(g)|.

g∈G

80

Analysis Block B

AB1 Limits

1 Limits of functions

1 A punctured neighbourhood of a point c is a bounded open interval with midpoint c, from which the point c itself has been removed:

Nr (c) = (c − r, c) ∪ (c, c + r), where r > 0.

2 Let f be a function defined on a punctured neighbourhood Nr (c) of c. Then f (x) tends to the limit l as x tends to c if

for each sequence xn in Nr(c) such that xn → c,

f (xn) → l.

In this case, we write

lim f (x) = l or f (x) → l as x → c. x→c

Strategy 1.1 To show that lim f (x) does x→c

not exist. either 1. find two sequences xn and yn which

tend to c, but whose terms are not equal to c, such that f (xn) and f (yn) have different limits;

or 2. find a sequence xn which tends to c, but

whose terms are not equal to c, such that f (xn) →∞ or f (xn) → −∞.

3 Evaluating limits

Theorem 1.2 Let f be a function defined on an open interval I, with c ∈ I. Then

f is continuous at c

if and only if lim f (x) = f (c). x→c

AB1

Combination Rules If lim f (x) = l and x→c

lim g(x) = m, then: x→c

Sum Rule lim(f (x) + g(x)) = l + m; x→c

Multiple Rule lim λf (x) = λl, for λ ∈ R; x→c

Product Rule lim f (x)g(x) = lm; x→c

Quotient Rule lim f (x)/g(x) = l/m, x→c provided that m = 0.

Composition Rule If lim f (x) = l and x→c

lim g(x) = L, then x→l

lim g(f (x)) = L, x→c

provided that either f (x) = l, for x in some Nr (c),

where r > 0, or g is defined at l and continuous at l.

Strategy 1.2 To use the Composition Rule. To evaluate a limit of a function of the form g(f (x)), as x → c: 1. substitute u = f (x) and show that, for

some l,u = f (x) → l as x → c;

2. show that, for some L, g(u) → L as u → l;

3. deduce that g(f (x)) → L as x → c.

Squeeze Rule Let f , g and h be functions defined on Nr (c), for some r > 0. If (a) g(x) ≤ f (x) ≤ h(x), for x ∈ Nr (c), (b) lim g(x) = lim h(x) = l,

x→c x→c

then

lim f (x) = l. x→c

81

AB1

Theorem 1.3 Three basic limits sin x

(a) lim = 1, x→0 x

1 − cos x(b) lim = 0,

x→0 xex − 1

(c) lim = 1. x→0 x

4 Let f be a function defined on (c, c + r), for some r > 0. Then f(x) tends to the limit l as x tends to c from the right if

for each sequence xn in (c, c + r) such that xn → c,

f(xn) → l.

In this case, we write

lim f(x) = l or f(x) → l as x → c + . +x→c

There is a similar definition for a limit as x tends to c from the left, in which (c, c + r) is replaced by (c − r, c). In this case, we write

−lim f (x) = l or f(x) → l as x → c . −x→c

We also refer to lim f(x) and lim f(x) as right+ −x→c x→c

and left limits, respectively.

Theorem 1.4 Let the function f be defined on Nr (c), for some r > 0. Then

lim f(x) = l x→c

if and only iflim f(x) = lim f(x) = l.

+ −x→c x→c

Theorem 1.5 Let f be a function whose domain is an interval I with a finite left-hand endpoint c that lies in I. Then

f is continuous at c

if and only iflim f(x) = f(c).

+x→c

There is an analogous result for left limits.

2 Asymptotic behaviour of functions

1 Let the function f be defined on Nr (c), for some r > 0. Then f(x) tends to ∞ as x tends to c if

for each sequence xn in Nr (c) such that xn → c,

f(xn) →∞.

In this case, we write

f(x) →∞ as x → c.

The statements f(x) → −∞ as x → c,

f(x) →∞ (or −∞) as x → c + (or c −), are defined similarly, with ∞ replaced by −∞ and Nr (c) replaced by the open interval (c, c + r) or (c − r, c), where r > 0, as appropriate.

Reciprocal Rule If the function f satisfies (a) f(x) > 0 for x ∈ Nr (c), for some r > 0, (b) f(x) → 0 as x → c, then

1/f (x) →∞ as x → c.

Combination Rules If f(x) →∞ as x → c and g(x) →∞ as x → c, then: Sum Rule f(x) + g(x) →∞ as x → c; Multiple Rule λf (x) →∞ as x → c,

for λ ∈ R+; Product Rule f(x)g(x) →∞ as x → c.

2 Let the function f be defined on (R, ∞), for some real number R. Then f(x) tends to l as x tends to ∞ if

for each sequence xn in (R, ∞) such that xn →∞,

f(xn) → l.

(Here l represents a real number or one of the symbols ±∞.) In this case, we write

f(x) → l as x →∞.

The statement f(x) → l as x → −∞

is defined similarly, with ∞ replaced by −∞, and (R, ∞) replaced by (−∞, R). When l is a real number, we also use the notations

lim f(x) = l and lim f(x) = l. x→∞ x→−∞

Theorem 2.1Basic asymptotic behaviourIf n ∈ N, then (a) xn →∞ as x →∞,

1(b) → 0 as x →∞.

xn

82

Squeeze Rule Let f , g and h be functions defined on some interval (R,∞). (a) If

1. g(x) ≤ f(x) ≤ h(x), for x ∈ (R,∞), 2. lim g(x) = lim h(x) = l,

x→∞ x→∞

where l is a real number, then

lim f(x) = l.x→∞

(b) If 1. f(x) ≥ g(x), for x ∈ (R,∞), 2. g(x) →∞ as x →∞, then

f(x) →∞ as x →∞.

Theorem 2.2 (a) If a0, a1, . . . , an−1 ∈ R, where n ∈ N, and

p(x) = x n + an−1x n−1 + · · ·+ a1x + a0,

then

p(x) →∞ as x →∞

and1 → 0 as x →∞.

p(x) (b) For each n = 0, 1, 2, . . . , we have

xe →∞ as x →∞ xn

and nx → 0 as x →∞.

ex

(c) We have

loge x →∞ as x →∞,

but, for each constant a > 0, we have loge x → 0 as x →∞.

xa

3 Continuity—the classical definition

1 Let the function f have domain A and let c ∈ A. Then f is continuous at c if

for each ε > 0, there exists δ > 0 such that |f(x) − f(c)| < ε, for all x ∈ A with |x− c| < δ.

AB1

Strategy 3.1 To use the ε–δ definition to prove continuity at a point.Let the function f have domain A, with c ∈ A.To prove that f is continuous at c, let ε > 0 be given and carry out the following.1. Use algebraic manipulation to express the

difference f(x) − f(c) as a product of the form (x− c)g(x).

2. Obtain an upper bound of the form |g(x)| ≤M , for |x− c| ≤ r, where r > 0 is chosen so that [c− r, c + r] ⊂ A.

3. Use the fact that |f(x) − f(c)| ≤M |x− c|, for |x− c| ≤ r, to choose δ > 0 such that |f(x) − f(c)| < ε, for all x ∈ A with |x− c| < δ.

Theorem 3.1 The ε–δ definition and the sequential definition of continuity are equivalent.

2 The Dirichlet function has domain R and rule 1, if x is rational,

f(x) = 0, if x is irrational.

Theorem 3.2 The Dirichlet function is discontinuous at every point of R.

3 The Riemann function has domain R and rule 1/q, if x is a rational p/q (q > 0),

f(x) = 0, if x is irrational.

Theorem 3.3 The Riemann function is discontinuous at each rational point of R and continuous at each irrational point.

4 The sawtooth function is defined by

x− [x], if 0 ≤ x− [x] ≤ 1

s(x) = 2 , 11 − (x− [x]), if 2 < x− [x] < 1,

where [x] is the integer part function. The blancmange function B is defined by

B(x) = s(x) + 1 s(2x) + 1 s(4x) + 1 s(8x) + · · · 2 4 8 ∞∑ 1

= s(2n x).2n

n=0

83

AB1

Theorem 3.4 The blancmange function is continuous.

5 Let f be a function defined on a punctured neighbourhood Nr (c) of c. Then f(x) tends to the limit l as x tends to c if

for each ε > 0, there exists δ > 0 such that |f(x) − l| < ε, for all x with 0 < |x− c| < δ.

As before, we write

lim f(x) = l or f(x) → l as x → c. x→c

4 Uniform continuity

1 A function f defined on an interval I is uniformly continuous on I if

for each ε > 0, there exists δ > 0 such that |f(x) − f(y)| < ε, for all x, y ∈ I with |x− y| < δ.

2 We say that c is an interior point of an interval I if c is not an endpoint of I.

3 Checking uniform continuity

Theorem 4.1 Let the function f be defined on an interval I. Then f is not uniformly continuous on I if and only if there exist two sequences xn and yn in I, and ε > 0, such that (a) |xn − yn| → 0 as n →∞, (b) |f(xn) − f(yn)| ≥ ε, for n = 1, 2, . . . .

Strategy 4.1 To check uniform continuity. Let the function f be defined on an interval I. 1. To prove that f is uniformly continuous

on I, find an expression for δ > 0 in terms of a given ε > 0 such that |f(x) − f(y)| < ε, for all x, y ∈ I with |x− y| < δ.

2. To prove that f is not uniformly continuous on I, find two sequences xn and ynin I, and ε > 0, such that |xn − yn| → 0 as n →∞,

|f(xn) − f(yn)| ≥ ε, for n = 1, 2, . . . .

Theorem 4.2 If the function f is continuous on a bounded closed interval [a, b], then f is uniformly continuous on [a, b].

Theorem 4.3 Bolzano–Weierstrass Theorem Any bounded sequence has a convergent subsequence.

84

AB2 Differentiation

1 Differentiable functions

1 Let f be defined on an open interval I, and c ∈ I. Then the difference quotient for f at c is

f (x) − f (c) f (c + h) − f (c) , or Q(h) = ,

x − c h= c, h where x = 0.

The slope, or gradient, of the graph of f at the point (c, f (c)) is

f (x) − f (c)lim , or lim Q(h), x→c x − c h→0

provided that the limit exists.

2 Let f be defined on an open interval I, and c ∈ I. Then the derivative of f at c is

f (x) − f (c)lim , x→c x − c

that is, f (c + h) − f (c)

lim Q(h) = lim , h→0 h→0 h

provided that this limit exists. In this case, we saythat f is differentiable at c.If f is differentiable at each point of its domain, thenwe say that f is differentiable (on its domain).The derivative of f at c is denoted by f ′(c), and thefunction f ′ : x −→ f ′(x) is called the derivative (or derived function) of f .

In Leibniz notation, f ′(x) is written as dy

, where y = f (x). dx

The operation of obtaining f ′(x) from f (x) is called differentiation.

Strategy 1.1 To prove that a function is not differentiable at a point, using the definition. Show that lim Q(h) does not exist, by:

h→0

either 1. finding two null sequences hn and h′

nwith non-zero terms such that the sequences Q(hn) and Q(h′

n) have different limits; or 2. finding a null sequence hn with non-zero

terms such that Q(hn) →∞ or Q(hn) → −∞.

AB2

Theorem 1.1 Basic derivatives (a) If f (x) = k, where k ∈ R, then f ′(x) = 0. (b) If f (x) = xn, where n ∈ N, then

n−1f ′(x) = nx . (c) If f (x) = sin x, then f ′(x) = cos x. (d) If f (x) = cos x, then f ′(x) = − sin x.

x(e) If f (x) = ex, then f ′(x) = e .

3 Let f be differentiable on an open interval I, and c ∈ I. If the derivative f ′ is differentiable at c, then we say that f is twice differentiable at c, and the number f ′′(c) = (f ′)′(c) is called the second derivative of f at c. The function f ′′, also denoted by f (2), is called the second derivative (or second derived function) of f .

d2yIn Leibniz notation, f ′′ is written as , where

dx2

y = f (x). Similarly, we can define the higher-order

= f ′′′derivatives of f , denoted by f (3) , f (4), and so on.

4 Let f be defined on an interval I, and c ∈ I. Then the left derivative of f at c is

f ′ f (x) − f (c)

L(c) = lim = lim Q(h), x→c− x − c h→0−

provided that this limit exists. In this case, we saythat f is left differentiable at c.Similarly, the right derivative of f at c is

f ′ f (x) − f (c)

R(c) = lim = lim Q(h), x→c+ x − c h→0+

provided that this limit exists. In this case, we say that f is right differentiable at c.

Theorem 1.2 Let f be defined on an open interval I, and c ∈ I. (a) If f is differentiable at c, then f is both left

differentiable and right differentiable at c, and

f ′ R(c) = f ′(c). (∗)L(c) = f ′

(b) If f is both left differentiable and right differentiable at c, and f ′ R(c), then L(c) = f ′

f is differentiable at c and equation (∗) holds.

85

( )′

AB2

Glue Rule Let f be defined on an open interval I, and c ∈ I. If there are functions g and h defined on I such that 1. f(x) = g(x), for x ∈ I, x < c,

f(x) = h(x), for x ∈ I, x > c,2. f(c) = g(c) = h(c), 3. g and h are differentiable at c,

then f is differentiable at c if and only if g′(c) = h′(c).If f is differentiable at c, then f ′(c) = g′(c) = h′(c).

5 Differentiability at a point is a local property ; it depends on the values of the function in any open interval (no matter how short) containing the point.

Corollary Let n p(x) = a0 + a1x + a2x 2 + · · ·+ anx ,

where a0, a1, . . . , an ∈ R. Then p is differentiable on R, with derivative

n−1 p ′(x) = a1 + 2a2x + · · ·+ nanx .

Composition Rule Let f be defined on an open interval I, let g be defined on an open interval J such that f(I) ⊆ J and let c ∈ I. If f is differentiable at c and g is differentiable at f(c), then g f is differentiable at c and

(g f)′(c) = g ′(f(c))f ′(c).

The restriction of a differentiable function to an open The Composition Rule for differentiation is often called the Chain Rule.interval gives a new differentiable function.

6 Continuity and differentiability

Theorem 1.3 Let f be defined on an open interval I, and c ∈ I. If f is differentiable at c, then f is continuous at c.

Corollary Let f be defined on an open interval I, and c ∈ I. If f is discontinuous at c, then f is not differentiable at c.

Theorem 1.4 The blancmange function B is not differentiable at any point of R.

2 Rules for differentiation

Combination Rules Let f and g be defined on an open interval I, and c ∈ I. If f and g are differentiable at c, then so are the functions: Sum Rule f + g, and

(f + g)′(c) = f ′(c) + g′(c); Multiple Rule λf , for λ ∈ R, and

(λf)′(c) = λf ′(c); Product Rule fg,

and (fg)′(c) = f ′(c)g(c) + f(c)g′(c);Quotient Rule f/g, provided that g(c) = 0,

f ′(c)and (c) =

g(c)f ′(c) − f(c)g.

g (g(c))2

86

Inverse Function Rule Let f be a function whose domain is an open interval I on which f is continuous and strictly monotonic, with image J = f(I). If f is differentiable on I and f ′(x) = 0 for x ∈ I, then f−1 is differentiable on its domain J . Also, if c ∈ I and d = f(c), then

1(f−1)′(d) =

f ′(c) .

3 Rolle’s Theorem

1 Let f be defined on an interval [a, b]. Then: f(d) is the maximum of f on [a, b] if d ∈ [a, b] and

f(x) ≤ f(d) for x ∈ [a, b]; f(c) is the minimum of f on [a, b] if c ∈ [a, b] and

f(x) ≥ f(c) for x ∈ [a, b]. An extremum is a maximum or a minimum.

2 The function f has

(a) a local maximum f(c) at c if there is an open interval I = (c− r, c + r), where r > 0, in the domain of f such that

f(x) ≤ f(c), for x ∈ I; (b) a local minimum f(c) at c if there is an open

interval I = (c− r, c + r), where r > 0, in the domain of f such that

f(x) ≥ f(c), for x ∈ I; (c) a local extremum f(c) at c if f(c) is either a

local maximum or a local minimum.

AB2

3 A point c such that f ′(c) = 0 is called a stationary point of f .

Theorem 3.1 Local Extremum Theorem If f has a local extremum at c and f isdifferentiable at c, then f ′(c) = 0.

Corollary Let f be continuous on the closed interval [a, b] and differentiable on (a, b). Then the extrema of f on [a, b] can occur only at a or b, or at points x in (a, b) where f ′(x) = 0.

Strategy 3.1 To find the maximum and minimum of a function. Let the function f be continuous on [a, b] and differentiable on (a, b). To determine the maximum and the minimum of f on [a, b]:

1. determine the points c1, c2, . . . in (a, b)where f ′ is zero;

2. amongst the values off(a), f(b), f(c1), f(c2), . . . ,

the greatest is the maximum and the least is the minimum.

Theorem 3.2 Rolle’s Theorem Let f be continuous on the closed interval [a, b] and differentiable on (a, b). If f(a) = f(b), then there exists a point c, with a < c < b, such that

f ′(c) = 0.

4 Mean Value Theorem

1 Mean Value Theorem

Theorem 4.1 Mean Value Theorem Let f be continuous on the closed interval [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that

f ′(c) = f(b) − f(a)

. b− a

2 For any interval I, the interior of I, denoted by Int(I), is the largest open subinterval of I.

Theorem 4.2 Increasing–Decreasing Theorem Let f be continuous on an interval I and differentiable on Int(I). (a) If f ′(x) ≥ 0 for x ∈ Int(I), then f is

increasing on I. (b) If f ′(x) ≤ 0 for x ∈ Int(I), then f is

decreasing on I.

Corollary Zero Derivative Theorem Let f be continuous on an interval I and differentiable on Int(I). If

f ′(x) = 0, for x ∈ Int(I), then

f is constant on I.

Second Derivative Test Let f be a twice differentiable function defined on an open interval containing a point c for which f ′(c) = 0 and f ′′ is continuous at c.

(a) If f ′′(c) > 0, then f(c) is a local minimum of f .

(b) If f ′′(c) < 0, then f(c) is a local maximum of f .

87

∑

∑

′

AB3

Strategy 4.1 To prove an inequality.To prove that g(x) ≥ h(x), for x ∈ [a, b], carryout the following.

1. Letf(x) = g(x) − h(x),

and show that f is continuous on [a, b] and differentiable on (a, b).

2. Prove either f(a) ≥ 0 and f ′(x) ≥ 0, for x ∈ (a, b), or f(b) ≥ 0 and f ′(x) ≤ 0, for x ∈ (a, b).

5 L’Hopital’s Rule

Theorem 5.1 The tangent to the curve with parametric equations

x = g(t), y = f(t),at the point with parameter t has slopef ′(t)/g′(t), provided that g′(t) = 0.

Theorem 5.2Cauchy’s Mean Value TheoremLet f and g be continuous on [a, b] and differentiable on (a, b). Then there exists apoint c ∈ (a, b) such that

f ′(c)(g(b) − g(a)) = g ′(c)(f(b) − f(a)); = g(a) and g′(c) in particular, if g(b) = 0, then

f ′(c) f(b) − f(a)=

g′(c) g(b) − g(a) .

L’Hopital’s Rule Let f and g be differentiable on an open interval I containing c, and suppose that f(c) = g(c) = 0. Then

f(x) f ′(x)lim exists and equals lim x→c g(x) x→c g′(x)

,

provided that the latter limit exists.

AB3 Integration

1 Riemann integral

1 Two important sums n

1 + 2 + 3 + · · ·+ n = i = 1 n(n + 1); 2 i=1

n

212 + 22 + 32 + · · ·+ n = i2 = 1 n(n + 1)(2n + 1). 6 i=1

2 Let f be defined on [a, b]. Then the followinghold on [a, b].min f = m if1. f(x) ≥ m for all x ∈ [a, b], 2. f(c) = m for some c ∈ [a, b].

max f = M if 1. f(x) ≤ M for all x ∈ [a, b], 2. f(d) = M for some d ∈ [a, b].

inf f = m if 1. f(x) ≥ m for all x ∈ [a, b], 2. if m > m, then f(c) < m′ for some c ∈ [a, b].

sup f = M if 1. f(x) ≤ M for all x ∈ [a, b], 2. if M ′ < M , then f(d) > M ′ for some d ∈ [a, b].

3 A partition P of an interval [a, b] is a collection of subintervals

P = [x0, x1], . . . , [xi−1, xi], . . . , [xn−1, xn], where

a = x0 < x1 < x2 < · · · < xn−1 < xn = b.

The length of the ith subinterval is denoted by

δxi = xi − xi−1.

The mesh of P is defined as ‖P‖ = max

1≤i≤nδxi.

A standard partition is a partition with subintervals of equal length.

4 If f is a bounded function defined on [a, b], and

P = [x0, x1], [x1, x2], . . . , [xn−1, xn], is a partition of [a, b], then, for i = 1, 2, . . . , n, we set

mi = inff(x) : xi−1 ≤ x ≤ xiand

Mi = supf(x) : xi−1 ≤ x ≤ xi.

88

∑

∑

∫

∫

The lower Riemann sum of f corresponding to P is

n

L(f, P ) = miδxi.i=1

The upper Riemann sum of f corresponding to P is

n

U(f, P ) = Miδxi.i=1

Theorem 1.1 If f is a bounded function on [a, b], and both P and P ′ are partitions of [a, b], then

L(f, P ) ≤ U(f, P ′).

5 Let f be bounded on [a, b]. The lower integral of f is ∫ b

f = sup L(f, P ),−a P

and the upper integral of f is ∫−b

f = inf U(f, P ),P a

where a and b are called limits of integration. We say that f is integrable on [a, b] if these lower ∫ b

and upper integrals are equal. The integral f is a

then defined to be the common value of the lower and upper integrals of f .

6 If P is a partition of [a, b], then any partition obtained from P by adding a finite number of points is called a refinement of P . The partition obtained from two partitions P and P ′ by using all their partition points is called the common refinement of P and P ′ .

Theorem 1.2 If f is an integrable function on [a, b] and Pn is a sequence of partitions of [a, b] such that ‖Pn‖ → 0, then ∫ b

lim L(f, Pn) = lim U(f, Pn) = f. n→∞ n→∞ a

Theorem 1.3 Let f be a bounded function on [a, b]. If there is a sequence of partitions Pn

of [a, b] such that ‖Pn‖ → 0 and

lim L(f, Pn) = lim U(f, Pn) = I, n→∞ n→∞

where I ∈ R, then f is integrable on [a, b] and ∫ b

f = I. a

AB3

Corollary Riemann’s Criterion Let f be bounded on [a, b]. Then

f is integrable on [a, b] if and only if

there is a sequence Pn of partitions of [a, b] with ‖Pn‖ → 0 such that U(f, Pn) − L(f, Pn) → 0.

Strategy 1.1 Determining integrability. Let f be bounded on [a, b]. 1. Choose any sequence of partitions Pn,

with ‖Pn‖ → 0. 2. Find L = lim L(f, Pn) and

n→∞U = lim U(f, Pn).

n→∞

If L = U , then f is not integrable.If L = U , then f is integrable, and∫ b

f = L = U. a

7 Limits of integration We define a

f = 0. a

a

If a > b and f exists, then we define b ∫ b ∫ a

f = − f. a b ∫ c ∫ b

8 Additivity of integrals If f and f a c

exist, then ∫ b ∫ b ∫ c ∫ b

f exists, and f = f + f. a a a c

If f is integrable on an interval I, then f is integrable on any subinterval of I, and ∫ b ∫ c ∫ b

f = f + f, for any a, b, c ∈ I. a a c

Sign of an integral Let f be integrable on [a, b]. ∫ b

If f(x) ≥ 0 on [a, b], then f ≥ 0. a ∫ b

If f(x) ≤ 0 on [a, b], then f ≤ 0. a

89

∫

∫

∫ ∫

∫

AB3

Modulus Rule If f is integrable on [a, b], then so is | f | .

Combination Rules If f and g areintegrable on [a, b], then so are:Sum Rule f + g, and ∫ b ∫ b ∫ b

(f + g) = f + g; a a a

Multiple Rule λf , for λ ∈ R, and ∫ b ∫ b

λf = λ f ; a a

Product Rule fg;

Quotient Rule f/g, provided 1/g is bounded on [a, b].

Theorem 1.4 A function f which is bounded and monotonic on [a, b] is integrable on [a, b].

Theorem 1.5 A function f which is continuous on [a, b] is integrable on [a, b].

2 Evaluation of integrals

1 Let f be a function defined on an interval I. Then a function F is a primitive of f on I if F is differentiable on I and

F ′(x) = f (x), for x ∈ I.

We can denote a primitive of f by f (x) dx.

Theorem 2.1Fundamental Theorem of CalculusLet f be integrable on [a, b], and let F be a primitive of f on [a, b]. Then ∫ b

f = F (b) − F (a). a

Often F (b) − F (a) is written as b[F (x)]b or F (x)| .a a

The process of finding a primitive of f is informally called integrating f , and in this context the function f is called an integrand.

Theorem 2.2 Uniqueness Theorem for Primitives Let F1 and F2 be primitives of f on an interval I. Then there exists some constant c such that

F2(x) = F1(x) + c, for x ∈ I.

Combination Rules Let F and G be primitives of f and g, respectively, on an interval I, and λ ∈ R. Then, on I: Sum Rule f + g has a primitive F + G; Multiple Rule λf has a primitive λF ; Scaling Rule x −→ f (λx) has a primitive

1 x −→ F (λx).

λ

2 Techniques of integration

Strategy 2.1 To find a primitive

f (g(x))g ′(x) dx, using integration by

substitution. du

1. Choose u = g(x); find = g′(x) and dx

express du in terms of x and dx. 2. Substitute u = g(x) and replace g′(x) dx by

du (adjusting constants if necessary) to give f (u) du.

3. Find f (u) du. 4. Substitute u = g(x) to give the required

primitive.

If we are evaluating an integral, rather than finding a primitive, then there is no need to perform step 4. Instead, we can change the x-limits of integration into the corresponding u-limits. If g(x) > 0 for x ∈ I, then

g′(x) dx = loge(g(x)).

g(x)

90

∫

∫

∫ ∫

( )

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣∣ ∣ ∣ ∣ ∣

∑ ∫

∑ ∫

∑

AB3

3 Inequalities, sequences and series

Strategy 2.2 To find a primitive f(x) dx,

using integration by backwards substitution. 1. Choose u = g(x), where g has inverse

function x = h(u), and express dx in terms of u and du.

2. Substitute x = h(u) and replace dx by h′(u) du to give a primitive in terms of u.

3. Find this primitive. 4. Substitute u = g(x) to give the required

primitive.

1 Inequalities for integrals

Inequality Rules Let f and g be integrable on [a, b]. (a) If f(x) ≤ g(x), for x ∈ [a, b], then ∫ b ∫ b

f ≤ g. a a

(b) If m ≤ f(x) ≤ M , for x ∈ [a, b], then ∫ b

m(b − a) ≤ f ≤ M(b − a). a

Strategy 2.3 To find a primitive k(x) dx, Triangle Inequality Let f be integrable[a, b]. Then ∫ b

using integration by parts. on

1. Write the original function k in the form fg′, where f is a function that you can

∫ b

f ≤ |f |a

. a

differentiate and g′ is a function that you can integrate.

Furthermore, if ∫ b

|f(x)| ≤ M for x ∈ [a, b], then

2. Use the formula

fg′ = fg − f ′ ≤ M(b − a).f

a g.

2 Formulas for π 3 To evaluate an integral In that involves a non-negative integer n, relate In to In−1 or In−2 by a reduction formula, using integration by parts. For example, let ∫ π/2

In = sinn x dx, n = 0, 1, 2, . . . . 0

Wallis’ Formula 2 2 4 4 2n 2n π

n→∞ 1 3 3 5 2n − 1 2n + 1(a) lim · · · · · · · · · = .

(n!)2 22n √ (b) lim √ = π.

n→∞ (2n)! nThen, using integration by parts twice, we obtain

n − 1 for n ≥ 2.In = In−2, 3 Integrals and series n

Also, I0 = π/2 and I1 = 1.

Hence Integral Test Let the function f be positive 1 3 5 2n − 1 π and decreasing on [1, ∞), and suppose that · · · · · · · ·I2n =

2 ,

2 4 6 2n f(x) → 0 as x →∞. Then 2 4 6 2n ∞

(a) f(n) converges if the sequence I2n+1 = · · · · · · · . n is3 5 7 2n + 1 f

1 n=1 bounded above; ∞

(b) f(n) diverges if n 1 f →∞ as n →∞.

n=1

Here, 1 can be replaced by any positive integer. ∞

4 The series 1/np converges for p > 1, and n=1

diverges for 0 < p ≤ 1.

2

91

AB4

4 Stirling’s Formula

For positive functions f and g with domain N, we write

f (n) ∼ g(n) as n →∞

to mean f (n) → 1 as n →∞. g(n)

Combination Rules If f1(n) ∼ g1(n) and f2(n) ∼ g2(n), then: Sum Rule f1(n) + f2(n) ∼ g1(n) + g2(n); Multiple Rule λf1(n) ∼ λg1(n), for λ ∈ R+; Product Rule f1(n)f2(n) ∼ g1(n)g2(n);

f1(n) g1(n)Quotient Rule ∼ .

f2(n) g2(n)

Stirling’s Formula √n! ∼ 2πn (n/e)n as n →∞.

AB4 Power series

1 Taylor polynomials

1 Let f be differentiable on an open interval containing the point a. Then the tangent approximation to f at a is

f (x) f (a) + f ′(a)(x − a).

2 Let f be n-times differentiable on an open interval containing the point a. Then the Taylor polynomial of degree n at a for f is the polynomial

Tn(x) = f (a) + f ′(a)(x − a) + f ′′(a)

(x − a)2 + · · · 2!

f (n)(a)+ (x − a)n .

n!

2 Taylor’s Theorem

1 Approximating by a Taylor polynomial

Theorem 2.1 Taylor’s Theorem Let the function f be (n + 1)-times differentiable on an open interval containing the points a and x. Then

f (x) = f (a) + f ′(a)(x − a) + · · · f (n)(a)

+ (x − a)n + Rn(x), n!

wheref (n+1)(c)

Rn(x) = (n + 1)!

(x − a)n+1 ,

for some c between a and x.

Taylor’s Theorem can be expressed in the form

f (x) = Tn(x) + Rn(x), where Rn(x) is a remainder term, or error term.

Strategy 2.1 To apply Taylor’s Theorem at a point and to show that the Taylor polynomial Tn at a for f approximates f to a certain accuracy at a point x = a, carry out thefollowing steps.1. Obtain a formula for f (n+1). 2. Determine a number M such that

|f (n+1)(c)| ≤M, for all c between a and x.

3. Write down and simplify the remainderestimate

M |Rn(x)| ≤ (n + 1)!

|x − a|n+1 .

92

∣ ∣ ∣ ∣∣ ∣ ∣ ∣

AB4

polynomial Tn at a for f approximates f to a ∞∑ certain accuracy on an interval I of the form [a, a + r], [a− r, a] or [a− r, a + r], where r > 0, carry out the following steps. 1. Obtain a formula for f (n+1). 2. Determine a number M such that

For a given power series an(x− a)n, exactly n=0

one of the following possibilities occurs.

(a) The series converges only for x = a.

(b) The series converges for all x. |f (n+1)(c)| ≤ M, for all c ∈ I. (c)

Write down and simplify the remainder There is a number R > 0 such that

∞∑3. estimate an(x− a)n converges if |x− a| < R

n=0M n+1 n(x)| ≤ for all x ∈ I.|R r , and(n + 1)!

∞∑ an(x− a)n diverges if |x− a| > R.

n=0Often the maximum value of |f (n+1)(c)| is takenwhen c is an endpoint of the interval.

2 Taylor series The positive number R appearing in case (c) above is

Let f have derivatives of all orders at the point a. called the radius of convergence of the power

The Taylor series at a for f is series.

f (n)(a) We write R = 0 if the power series converges only for (x− a)n = f(a) + f ′(a)(x− a) x = a, and R = ∞ if the power series converges for

n! all x.

∞∑

n=0 f ′′(a)

+ (x− a)2 + · · · . The interval of convergence of a power series is 2! the interval (a−R, a + R), together with any endpoints of this interval at which the power series

Theorem 2.2 Let f have derivatives of all converges. orders on an open interval containing the points a and x. If

∞∑Rn(x) → 0 as n →∞, Ratio Test Suppose that an(x− a)n is a

then n=0 ∞∑ power series with radius of convergence R, and

an+1

f (n)(a)f(x) = (x− a)n . (∗)

n! → L as n →∞n=0 . an

(a) If L is ∞, then R = 0.

(b) If L = 0, then R = ∞. (c) If L > 0, then R = 1/L.

Strategy 3.1 To find the interval of

If x is a point for which the Taylor series for f has sum f(x), as in equation (∗), then we say that the Taylor series is valid at the point x. Any set of values of x for which a Taylor series is valid is called a range of validity for the Taylor series. On any such range of validity, the function f is the sum function of the Taylor series. ∞∑

convergence of an(x− a)n . n=0

3 Convergence of power series 1. Use the Ratio Test for power series to find the radius of convergence R.

Let a ∈ R, x ∈ R and an ∈ R, n = 0, 1, 2, . . . . Then 2. Use other tests for convergence of series to the expression determine the behaviour of the power series

∞∑ at the endpoints of the interval an(x− a)n = a0 + a1(x− a) + a2(x− a)2 + · · · (a−R, a + R).

n=0

is called a power series about a in x, with coefficients an. We call a the centre of the power series.

Strategy 2.2 To apply Taylor’s Theorem on Theorem 3.1 an interval and to show that the Taylor Radius of Convergence Theorem

93

∫

( )

)

( )

( ) ( )

)

( ) ( ) ( )

c

AB4

4 Manipulating power series 2 The generalised binomial coefficients are

α = 1,1 Determining new power series 0

and( α α(α− 1)(α− 2) · · · (α− n + 1)

Combination Rules If = , n ∈ N. n n!∞∑

f(x) = an(x− a)n , for |x− a| < R, n=0 Theorem 4.1

< R′ , General Binomial Theorem ∞∑

g(x) = bn(x− a)n , for |x− a| For α ∈ R,n=0

then the following hold.

Sum Rule

∞∑ α(1 + x)α n , for |x| < 1.= x n

n=0∞∑ (f + g)(x) = (an + bn)(x− a)n ,

n=0

for |x− a| < r = minR, R′. Multiple Rule For λ ∈ R,

Theorem 4.2 If

Identity Theorem

∞∑∞∑∞∑ (λf)(x) = λan(x− a)n , for |x− a| < R. an(x− a)n = bn(x− a)n , for |x| < R,

n=0 n=0 n=0

thenProduct Rule

∞∑ an = bn, for n = 0, 1, . . . . (fg)(x) = cn(x− a)n ,

n=0

x− a , where Thus any valid method of obtaining the power series < r = minR, R′for | | ∞∑ n = a0bn + a1bn−1 + · · ·+ an n(x− a)n gives the same coefficients. b0. a

n=0

Differentiation Rule The power series 5 Numerical estimates for π ∞∑∞∑ an(x− a)n and nan(x− a)n−1

1 These formulas can be used to estimate π: n=0 n=1

have the same radius of convergence, R say. tan−12 1 + tan−1 1 = π/4, ( 3( )∞∑ 4 tan−1 − tan−11 1 = π/4 (Machin’s Formula),

Also, f(x) = an(x− a)n is differentiable on n=0

5 239

6 tan−1 + 2 tan−1 + tan−11 1 1 = π/4.8 57 239 (a−R, a + R), and

∞∑ f ′(x) = nan(x− a)n−1 , for |x− a| < R.

n=1

Integration Rule The power series

Theorem 5.1 The number π is irrational.

∞∑ f(x) = an(x− a)n

n=0

and ∞∑

F (x) = n=0

an

n + 1(x− a)n+1

have the same radius of convergence, R say. Also, if R > 0, then f is integrable on

(a−R, a + R), and f(x) dx = F (x).

94

Appendix

Sketches of graphs of basic functions

95

Sketches of graphs of standard inverse functions

96

Properties of trigonometric and hyperbolic functions

Trigonometric functions Hyperbolic functions

cos is even: cos(−x) = cos x cosh is even: cosh(−x) = cosh x sin is odd: sin(−x) = − sin x sinh is odd: sinh(−x) = − sinh x tan is odd: tan(−x) = − tan x tanh is odd: tanh(−x) = − tanh x

cos2 x + sin2 x = 1 cosh2 x − sinh2 x = 1

1 + tan2 x = sec2 x 1 − tanh2 x = sech2 x

cot2 x + 1 = cosec2 x coth2 x − 1 = cosech2 x

sin(x + y) = sin x cos y + cos x sin y sinh(x + y) = sinh x cosh y + cosh x sinh y cos(x + y) = cos x cos y − sin x sin y cosh(x + y) = cosh x cosh y + sinh x sinh y

tan(x + y) = tan x + tan y

1 − tan x tan y tanh(x + y) =

tanh x + tanh y 1 + tanh x tanh y

sin 2x = 2 sin x cos x sinh 2x = 2 sinh x cosh x

cos 2x = cos2 x − sin2 x cosh 2x = cosh2 x + sinh2 x = 2 cos2 x − 1 = 2 cosh2 x − 1 = 1 − 2 sin2 x = 1 + 2 sinh2 x

tan 2x = 2 tan x

1 − tan2 x tanh 2x =

2 tanh x

1 + tanh2 x

sin(π − x) = sin x cos(π − x) = − cos x

Some standard values of sin, cos and tan

These can be found from the following triangles.

97

)

x

Standard derivativesf (x) f ′(x) Domain

k 0 R x 1 R xn, n ∈ Z − 0xα, α ∈ R

nxn−1

αxα−1 R R+

ax, a > 0 ax loge a R

sin x cos x R cos x tan x

− sin x sec2 x

R R −

( n + 1

2

) π : n ∈ Z

cosec x sec x

− cosec x cot x sec x tan x

R − nπ : n ∈ ZR −

( n + 1

2

) π : n ∈ Z

cot x sin−1 x

− cosec2 x 1/ √

1 − x2 √

R − nπ : n ∈ Z(−1, 1)

cos−1 x tan−1 x

−1/ 1 − x2

1/ ( 1 + x2

) (−1, 1) R

e ex R loge x 1/x R+

sinh x cosh x R cosh x sinh x R tanh x sech2 x R√ sinh−1 x 1/√1 + x2 R cosh−1 x 1/( x

2 − 1 (1, ∞) tanh−1 x 1/ 1 − x2 (−1, 1)

Standard Taylor series Function Taylor series Domain

1 1 − x

1 + x + x2 + x3 + · · · = ∞∑

n=0

x n |x| < 1

loge(1 + x) x − x2

2 +

x3

3 −

x4

4 + · · · =

∞∑ (−1)n+1xn

n −1 < x ≤ 1

n=1

ex 1 + x + x2

2! +

x3

3! + · · · =

∞∑ xn

n! x ∈ R

n=0

sin x x − x3

3! +

x5

5! −

x7

7! + · · · =

∞∑

n=0

(−1)nx2n+1

(2n + 1)! x ∈ R

cos x

(1 + x)α

1 − x2

2! +

x4

4! −

x6

6! + · · · =

∞∑

n=0

(−1)nx2n

(2n)!

1 + αx + α(α − 1)

2! x2 +

α(α − 1)(α − 2) 3!

x3 + · · · = ∞∑

n=0

( α n

)

x n

x ∈ R

|x| < 1, α ∈ R

sinh x x + x3

3! +

x5

5! +

x7

7! + · · · =

∞∑

n=0

x2n+1

(2n + 1)! x ∈ R

cosh x 1 + x2

2! +

x4

4! +

x6

6! + · · · =

∞∑

n=0

x2n

(2n)! x ∈ R

tan−1 x x − x3

3 +

x5

5 −

x7

7 + · · · =

∞∑

n=0

(−1)nx2n+1

2n + 1 |x| ≤ 1

98

( )

Standard primitives

f (x) Primitive F (x) Domain

nx , n ∈ Z − −1 xn+1/(n + 1) R xα, α = −1 xα+1/(α + 1) R+

ax, a > 0 ax/ loge a R

sin x − cos x R cos x sin x R tan x loge(sec x) (− 1

2 π, 1 2 π)

xe1/x 1/x loge x sinh x cosh x tanh x

xeloge x loge |x|x loge x − x cosh x sinh x loge(cosh x)

R (0, ∞) (−∞, 0) (0, ∞) R R R

2)−1(a2 − x , a = 0

2)−1(a2 + x , a = 0

2)−1/2(a2 − x , a = 0

2)−1/2(x2 − a , a = 0

2)−1/2(a2 + x , a = 0

2)1/2(a2 − x , a = 0

2)1/2(x2 − a , a = 0

2)1/2(a2 + x , a = 0

eax cos bx, a, b = 0

eax sin bx, a, b = 0

1 a + xloge (−a, a)

2a a − x

1 tan−1(x/a) R

a

sin−1(x/a) (−a, a) − cos−1(x/a) (−a, a)

loge(x + (x2 − a2)1/2) (a, ∞) cosh−1(x/a) (a, ∞)

loge(x + (a2 + x2)1/2) R sinh−1(x/a) R

1 x(a2 − x2)1/2 + 1 a2 sin−1(x/a) (−a, a)2 2

2)1/2 − 11 x(x2 − a a2 loge(x + (x2 − a2)1/2) (a, ∞)2 2

1 x(a2 + x2)1/2 + 1 a2 loge(x + (a2 + x2)1/2) R2 2 axe

2 + b2 (a cos bx + b sin bx) R a

axe2 + b2 (a sin bx − b cos bx) R

a

99

Group tables of symmetry groups

e a b c r s t u

e e a b c r s t u a a b c e s t u r b b c e a t u r s c c e a b u r s t r r u t s e c b a s s r u t a e c b t t s r u b a e c u u t s r c b a e

e a b r s t

e e a b r s t a a b e t r s b b e a s t r r r s t e a b s s t r b e a t t r s a b e

Groups of small order

e a r s

e e a r s a a e s r r r s e a s s r a e

e a b c

e e a b c a a b c e b b c e a c c e a b

Order Number of Example of each class isomorphism

classes Abelian Non-Abelian

1

2

1

1

eC2

–

–

3 1 C3 –

4 2 C4, with 1 element of order 2 – K4, with 3 elements of order 2

5 1 C5 –

6 2 C6, with 1 element of order 2 S(), with 3 elements of order 2

7 1 C7 –

8 5 C8, with 1 element of order 2 S(), with 5 elements of order 2 A group with 7 elements of order 2 A group with 1 element of order 2 A group with 3 elements of order 2

100

Three types of non-degenerate conic

Six types of non-degenerate quadric

101

Index

Abelian group, 25, 29, 36, 37, 73, 76of order 8, 36

Absolute Convergence Test, 68absolute value, 61absolutely convergent series, 68action table, 78acts on, 78addition

in modular arithmetic, 21of complex numbers, 19of vectors, 40

additive group, 30, 37additive inverse, 18

in modular arithmetic, 21of a complex number, 19of a matrix, 44

additive notation, 28additivity of integrals, 89alternating group An, 33, 37Alternating Test, 68angle

between vectors, 41of rotation, 26

Antipodal Points Theorem, 71Archimedean Property, 60Argand diagram, 18argument of a complex number, 19associativity, 18, 60

in a group, 25in a vector space, 48of matrix addition, 44, 75of matrix multiplication, 44, 75

asymptote, 9, 10of a conic section, 41

asymptotic behaviour of functions, 82augmented matrix, 43automorphism, 75axioms of a group, 24axis

of rotation, 26of symmetry, 23, 26

basic continuous functions, 71basis, 49

for the image of a linear transformation, 54Basis Theorem, 50Bernoulli’s Inequality, 61bijection, 15binary operation, 24binomial coefficient, 14

generalised, 94Binomial Theorem, 17, 61bisection method, 71blancmange function, 83blocking a group table, 37, 38Bolzano–Weierstrass Theorem, 84bounded above, 62

bounded below, 62bounded figure, 23, 26bounded function, 71bounded sequence, 65bounded set, 62Boundedness Theorem, 71

cancellation laws, 26cardioid, 13Cartesian form, 19Cauchy’s Mean Value Theorem, 88Cayley table, 24Cayley’s Theorem, 34centre

of a conic section, 41of a power series, 93

Chain Rule, 86characteristic equation, 56circle, 14

equation of, 41parametrisation of, 12

closed interval, 9closed set, 23closure, 18, 60

in a vector space, 48of a group, 24

codomain, 9, 14 coefficient matrix, 45coefficients, 43

of a power series, 93cofactor, 47column matrix, 44Combination Rules

for ∼, 92 for continuity, 70for differentiation, 86for inequalities, 61for integrals, 90for limits, 64, 65, 81, 82for null sequences, 63for power series, 94for primitives, 90for series, 67

common factor, 21common refinement, 89commutative group, 25commutativity, 18, 23, 60

in a vector space, 48of matrix addition, 44

Comparison Test, 67completed-square form, 9complex conjugate, 19complex exponential function, 20complex number, 18complex plane, 18components of a vector, 40composite function, 12, 15

102

composite of transpositions, 32 counter-example, 16 composition Counting Theorem, 80

of functions, 23 cube, 26 of permutations, 31 cubic equation, 18 of symmetries, 24, 26 cubic function, 9

Composition Rule graph, 95 for continuity, 70 cycle, 31, 32 for differentiation, 86 cycle form, 31 for limits, 81 cycle structure, 32 for linear transformations, 53 cyclic group, 28–30, 36, 76

conclusion, 15 cyclic subgroup, 28, 30 congruence, 21 cycloid, 12 conics, 12, 41

non-degenerate, 101 de Moivre’s Theorem, 20 standard form, 58 decimal, 18, 60

conjugacy, 73 decreasing function, 10, 69 in symmetry groups, 74 decreasing sequence, 63 of permutations in Sn, 33–34 degenerate conic section, 41

conjugacy class, 73, 80 degree conjugate, complex, 19 of a permutation group, 32 conjugate group elements, 33, 73 of a polynomial, 11 conjugate subgroups, 34, 73 Density Property, 60 Conjugate Subgroups Theorem, 34 derivative, 85, 98 conjugating element, 33, 73 derived function, 85 consistent system, 43 determinant, 46 constant sequence, 63 diagonal matrix, 45 constant term, 43 diagonal of a matrix, 45 continuity diagonalisable matrix, 57

and differentiability, 86 difference and limits, 81 between sets, 14 classical definition, 83 between vectors, 40 sequential definition, 70 difference quotient, 85 uniform, 84 differentiable, 85

continuous at a point, 70 differentiation, 85 continuous function, 70, 83 Differentiation Rule for power series, 94

basic, 71 digit, 60 continuous on an interval, 71 dilation, 52 contradiction, 16 dimension, 50 contraposition, 16 Dimension Theorem, 55 contrapositive, 16 direct symmetry, 24, 26, 27, 75 convergent sequence, 64 directrix, 41 convergent series, 66, 91 Dirichlet function, 83 converse, 15 disc, 14 convex polyhedron, 26 symmetries of, 24 coordinates of a vector, 49 disjoint cycles, 31 coprime numbers, 21, 22, 30 disjoint sets, 14, 22 corollary, 15 Distance Formula, 19, 39 Correspondence Theorem, 77, 80 distributive law for matrices, 44 cosech function, 12 distributivity, 18, 60 coset, 35, 80 in a vector space, 48

in an additive group, 35, 37 divergent sequence, 65 left, 37 divergent series, 66, 91 of a stabiliser, 80 Division Algorithm, 21 of the kernel of a homomorphism, 77 division of complex numbers, 19 right, 37 divisor, 21

cosh function, 12 dodecahedron, 26 graph, 95, 96 domain, 9, 10, 14

cosine function, 97 dominant term, 11, 64 graph, 95, 96 dominated sequence, 64 Taylor series for, 98 dot product, 40

coth function, 12 in Rn, 50

103

eccentricity, 41E-coordinate representation, 49eigenspace, 56eigenvalue, 56eigenvector, 56eigenvector basis, 56, 57eigenvector equations, 56element of a set, 13elementary matrix, 46elementary operations, 43elementary row operation, 43

inverse, 46ellipse, 12, 41, 42, 101ellipsoid, 101elliptic cone, 101elliptic paraboloid, 101empty set, 13ε–δ definition of continuity, 83equality

of matrices, 44of vectors, 40

equivalence, 15equivalence class, 22equivalence relation, 22error term, 92Euclidean space, 39Euclid’s Algorithm, 22Euler’s formula, 20even function, 10even permutation, 33even subsequence, 65exhaustion, 16existential quantifier, 16existential statement, 16exponent laws, 62exponential (e), 66exponential form of a complex number, 20exponential function, 9, 12, 69, 72

complex, 20graph, 95, 96Taylor series for, 98

Exponential Inequalities, 71extreme value, 71Extreme Value Theorem, 71extremum of a function, 86

factorial, 14, 32field, 18, 60figure, 26finite decimal, 60finite dimension, 49finite group, 25, 28finite set, 14First Derivative Test, 10First Subsequence Rule, 65fixed point set, 74Fixed Point Theorem, 74fixed set, 80fixed symbol in a permutation, 31focal chord, 41focus, 41

fractional part, 38function, 9, 14Fundamental Theorem of Algebra, 20Fundamental Theorem of Arithmetic, 16Fundamental Theorem of Calculus, 90

Gauss–Jordan elimination, 44General Binomial Theorem, 94generalised binomial coefficients, 94generated set, 28generator of a cyclic subgroup, 28, 30Geometric Series Identity, 17geometric series, sum of, 17, 66geometric type of a symmetry, 73glide-reflection, 24Glue Rule

for continuity, 70for differentiation, 86

gradient, 85Gram–Schmidt orthogonalisation, 51graph of a real function, 14graph sketching, 10, 11greater than, 60greatest common factor, 21greatest lower bound, 62Greatest Lower Bound Property, 62Greek alphabet, 5group, 24

axioms, 24finite, 25infinite, 25of even order, 36of order 1, 36of order 4, 36of order 6, 36of order 8, 36of prime order, 36of small order, 36, 100

group action, 78table, 78

group table, 25, 100blocking of, 38

half-closed interval, 9half-open interval, 9half-plane, 13higher-order derivative, 85homogeneous system, 43homomorphism, 75homomorphism property, 75horizontal asymptote, 10horizontal point of inflection, 10hybrid function, 12hyperbola, 9, 41, 42, 101

parametrisation of, 13rectangular, 42

hyperbolic functions, 12, 97graphs, 95, 96

hyperbolic paraboloid, 101hyperboloid, 101hyperplane, 51hypothesis, 15

104

icosahedron, 26identifying up to isomorphism, 77identity, 17, 18, 23, 60

in a group, 24, 25in a quotient group, 38in a vector space, 48in C, 19

identity function, 14identity matrix, 45identity permutation, 31identity symmetry, 23Identity Theorem, 94identity transformation, 52image, 9, 14

of a homomorphism, 76of a linear transformation, 54

image set, 14imaginary axis, 18imaginary number, 18imaginary part, 18implication, 15, 16inconsistent system, 43increasing function, 10, 69increasing sequence, 63increasing/decreasing criterion, 10Increasing–Decreasing Theorem, 87index of a subgroup, 35indirect symmetry, 24, 26, 75induction, mathematical, 16inequalities, 60

exponential, 71for natural numbers, 61for real numbers, 61proving, 62rules, 61

Inequality Rules for integrals, 91infimum

of a function, 88of a set, 62

infinite decimal, 60infinite dimension, 49infinite group, 25, 28

cosets of, 35infinite series, 66integer, 13, 60integer part function, 9

graph, 95integrable, 89integral, 89Integral Test, 91integrand, 90integration

by backwards substitution, 91by parts, 91by substitution, 90

Integration Rule for power series, 94intercept, 10interior, 87interior point, 84Intermediate Value Theorem, 71intersection of sets, 14

interval, 9of convergence, 93

inverse, 18, 60in a group, 25, 26in a quotient group, 38in a vector space, 48of a matrix, 45, 47of a permutation, 32

inverse function, 15, 69graphs, 96

Inverse Function Rule, 72for derivatives, 86

inverse hyperbolic functions, 72graphs, 96

Inverse Rule, 54inverse symmetry, 23inverse trigonometric functions, 72

graphs, 96Invertibility Theorem, 45invertible linear transformation, 54invertible matrix, 45, 47irrational number, 18, 60isometry, 23, 26isomorphic groups, 29isomorphic linear transformations, 54isomorphism, 29, 75

of cyclic groups, 30of linear transformations, 54

isomorphism class, 29for groups of order 8 or less, 36

Isomorphism Theorem, 77

k-dilation, 52kernel

of a homomorphism, 76, 77of a linear transformation, 55

Klein group, 29, 36(k, l)-stretching, 52

Lagrange’s Theorem, 35, 80leading diagonal

of a Cayley table, 24of a matrix, 45

least common multiple, 32least upper bound, 62Least Upper Bound Property, 62left derivative, 85left limit, 82Leibniz notation, 85lemma, 15length

of a cycle, 32of a vector, 39, 40, 51

less than, 60l’Hˆopital’s Rule, 88Limit Comparison Test, 67Limit Inequality Rule, 64limit of a function, 81, 84

and continuity, 81does not exist, 81from the right or left, 82

limit of a sequence, 64

105

limits of integration, 89line, 13

equation of, 39graph, 95parametrisation of, 12vector form of equation, 40

linear combination of vectors, 48, 52linear equation, 18, 22

in n unknowns, 43linear function, 9linear independence, 49linear rational function, 9linear transformation, 52linearly dependent set, 49linearly independent set, 49local extremum of a function, 86Local Extremum Theorem, 87local maximum, 10

of a function, 86local minimum, 10

of a function, 86local property

of continuity, 70of differentiability, 86

logarithm function, 72graph, 95, 96Taylor series for, 98

lower bound, 62greatest, 62

lower integral, 89lower Riemann sum, 89lower-triangular matrix, 45, 75

magnitude of a vector, 39major axis of a hyperbola, 42many-one function, 15mapping, 14mathematical induction, 16matrix, 43

addition, 44multiplication, 44

matrix form of a system of equations, 45matrix group, 75, 79matrix representation of a linear transformation, 53maximum element, 62maximum of a function, 10, 86, 88maximum value, 71Mean Value Theorem, 87member of a set, 13mesh, 88minimal spanning set, 48minimum element, 62minimum of a function, 10, 86, 88minimum value, 71minor axis of a hyperbola, 42modular arithmetic, 21, 22, 30modulus, 61

in modular arithmetic, 21of a complex number, 19of a real number, 9

modulus function, 9

graph, 95 Modulus Rule

for integrals, 90Monotone Convergence Theorem, 66monotonic function, 69monotonic sequence, 63Monotonic Sequence Theorem, 66multiple of an element in an additive group, 28Multiple Rule

for ∼, 92 for continuity, 70for differentiation, 86for integrals, 90for limits, 64, 65, 81, 82for null sequences, 63for power series, 94for primitives, 90for series, 67

multiplicationin modular arithmetic, 21of complex numbers, 19

Multiplication Rule, 80multiplicative group, 30multiplicative inverse, 18, 21, 22

in modular arithmetic, 21of a complex number, 19

multiplicative notation, 28multiplicity of an eigenvalue, 56

natural number, 13, 60n-dimensional space, 48negation, 15, 16negative, 18

of a complex number, 19of a matrix, 44of a vector, 39, 40

negative angle, 23n-gon, 23non-Abelian group, 25

of order 8, 36non-cyclic group, 28non-degenerate conic, 41, 101non-degenerate quadric, 101non-homogeneous system, 43Non-null Test, 67non-terminating decimal, 60non-trivial solution, 43normal subgroup, 37, 74normal vector, 41nth partial sum, 66nth root, 62nth root function, 72nth term

of a sequence, 63of a series, 66

n-tuple, 48null sequence, 63number line, 60

octahedron, 26odd function, 10odd permutation, 33

106

odd subsequence, 65one-one correspondence, 15one-one function, 15onto function, 15open interval, 9orbit, 79Orbit–Stabiliser Theorem, 80order

infinite, 25, 28of a group, 25, 35, 36of a group element, 28, 73of a kernel, 77of a permutation, 32of a quotient group, 38of an image, 77prime, 36

order properties of R, 60 orthogonal basis, 50orthogonal matrix, 57, 58orthogonal set of vectors, 50orthogonal vectors, 41

in Rn, 50 orthogonalisation (Gram–Schmidt), 51orthogonally diagonalisable matrix, 57orthonormal basis, 51, 57orthonormal eigenvector basis, 58

parabola, 9, 41, 101graph, 95parametrisation of, 13

parallel lines, 39Parallelogram Law, 39parameter, 12parametric equations, 12parametrisation, 12, 14parity of a permutation, 33Parity Theorem, 33partial sum of a series, 66partition, 22

of a group into conjugacy classes, 73, 74of a group into cosets, 35, 37of a set into orbits, 79of an interval, 88

Pascal’s triangle, 17periodic function, 10permutation, 31permutation group, 32perpendicular lines, 39perpendicular vectors, 41pi (π), 66

estimating, 94formulas for, 91

plane, equation of, 39, 41plane figure, 13, 23plane of reflection, 26plane set, 13Platonic solid, 26point of inflection, 10polar form, 19polyhedron, 26polynomial, 17

polynomial equation, 18Polynomial Factorisation Theorem, 17, 20polynomial function, 11position vector, 40power function, graph, 96power of a group element, 28Power Rule, 63power series, 93, 98preservation of composites, 75prime number, 13primitive, 90, 99principal argument, 19Principle of Mathematical Induction, 16product

of disjoint cycles, 31of matrices, 44of real numbers, 62

Product Rulefor ∼, 92 for continuity, 70for differentiation, 86for inequalities, 61for integrals, 90for limits, 64, 65, 81, 82for null sequences, 63for power series, 94

projection of a vector, 41proof, 16

by contradiction, 16by contraposition, 16by exhaustion, 16by mathematical induction, 16

proper subgroup, 27proper subset, 14proposition, 15punctured neighbourhood, 81

quadratic equation, 18quadratic function, 9quadric, 58

non-degenerate, 101standard form, 59

quotient, 21of complex numbers, 19

quotient group, 38of an infinite group, 38

Quotient Rulefor ∼, 92 for continuity, 70for differentiation, 86for integrals, 90for limits, 64, 81

radius of convergence, 93Radius of Convergence Theorem, 93range of validity, 93Ratio Test

for power series, 93for series, 68

rational function, 11rational number, 13, 18, 60r-cycle, 32

107

real axis, 18 real function, 9real line, 18, 60real number, 13, 18, 60real part, 18real vector space, 48reciprocal, 18

of a complex number, 19reciprocal function, 9

graph, 95Reciprocal Rule, 65

for functions that tend to ∞, 82 rectangular hyperbola, 42recurring decimal, 60reduction formula, 91refinement, 89reflection, 23, 26, 52reflexive property, 22regular n-gon, 23regular polyhedron, 26relation, 22relatively prime, 21remainder, 21remainder term, 92‘renaming’ permutations, 33representative of an equivalence class, 22restriction of a function, 15, 70, 86Riemann function, 83Riemann sum, 89Riemann’s Criterion, 89right derivative, 85right limit, 82Rolle’s Theorem, 87root, 62

of a complex number, 20of a polynomial, 20of unity, 20

root function, graph, 96rotation, 23, 26, 52row matrix, 44row-reduced form of a matrix, 43rule for a function, 9, 14

sawtooth function, 83scalar, 39scalar multiple

of a matrix, 44of a vector, 39

scalar multiplication, 40, 48scaling, 9Scaling Rule, 90sech function, 12second derivative, 85Second Derivative Test, 11, 87second derived function, 85Second Subsequence Rule, 65Section Formula, 40self-conjugate subgroup, 73self-inverse element, 23, 26sequence, 63sequence diagram, 63

series, 66set, 13

finite, 14set composition, 37shear, 52sigma notation, 66sign of an integral, 89simultaneous linear equations, 55sine function

graph, 95, 96Taylor series for, 98

Sine Inequality, 70singleton, 13sinh function, 12

graph, 95, 96size of a matrix, 44slope, 85solution set, 13, 61

of a system of equations, 43Solution Set Theorem, 55span, 48spanning set, 48square matrix, 44, 58Squeeze Rule

for continuity, 70for limits, 64, 65, 81, 83for null sequences, 64

stabiliser, 79standard basis, 49standard form

of a conic, 58of a quadric, 59

standard partition, 88statement, 15stationary point, 10, 87Stirling’s Formula, 92straight line, 13stretching, 52strict inequality, 60, 61strictly decreasing function, 69strictly decreasing sequence, 63strictly increasing function, 69strictly increasing sequence, 63strictly monotonic function, 69strictly monotonic sequence, 63subgroup, 27

of symmetry group, 27possible order of, 35

submatrix, 47subsequence, 65Subsequence Rules, 65subset, 14subspace, 50, 56subtraction

of complex numbers, 19of matrices, 44of vectors, 40

sumof a series, 66of real numbers, 62

sum function, 93

108

Sum Rulefor ∼, 92 for continuity, 70for differentiation, 86for inequalities, 61for integrals, 90for limits, 64, 65, 81, 82for null sequences, 63for power series, 94for primitives, 90for series, 67

supremumof a function, 88of a set, 62

symmetric group S4, 32 symmetric group Sn, 32, 37symmetric matrix, 45, 58symmetric property, 22symmetry, 23, 26

equality of, 23, 26of a regular n-gon, 23of the disc, 23

symmetry group, 100subgroup, 27

system of equations in matrix form, 45system of linear equations, 43

tangent approximation, 92tangent function, graph, 95, 96tangent to a curve, 88tanh function, 12

graph, 95, 96Taylor polynomial, 92Taylor series, 93Taylor’s Theorem, 92techniques of integration, 90telescoping series, 67tends to infinity, 65, 82term

of a sequence, 63of a series, 66

terminating decimal, 60tetrahedron, 26, 33, 34theorem, 15three-dimensional Euclidean space, 39trace, 56transition matrix, 56Transitive Property, 60transitive property of an equivalence relation, 22Transitive Rule, 61translation, 9, 23, 52transpose of a matrix, 45

transposition, 32Triangle Inequality, 61

for integrals, 91infinite form, 68

Triangle Law, 39Trichotomy Property, 60trigonometric functions, 97

graphs, 95, 96trisectrix, 13trivial homomorphism, 75trivial solution, 43trivial subgroup, 27trivial symmetry, 23twice differentiable, 85two-dimensional Euclidean space, 39two-line symbol, 24, 31

unbounded sequence, 65unbounded set, 62uniform continuity, 84union of sets, 14Uniqueness Theorem for Primitives, 90unit circle, 13unit vector, 40universal quantifier, 16universal statement, 16upper bound, 62

least, 62upper integral, 89upper Riemann sum, 89upper-triangular matrix, 45, 75

valid, 93vanish, 71vector, 39vector addition, 48vector space, 40Venn diagram, 13vertical asymptote, 10

Wallis’ Formula, 91weak inequality, 60, 61wedge symbol, 78

zeroin a vector space, 48of a function, 10, 71of a polynomial, 20of a polynomial function, 71

zero complex number, 18Zero Derivative Theorem, 87zero matrix, 44zero transformation, 52zero vector, 39, 49

109