sample linear algebra exams

7/25/2019 Sample Linear Algebra Exams

1/10

Math10

F irst

Exam

Instructions:

Show complete answers to the following questions in your bluebook. An answer

without a solution, even if correct,willget little or no credit. The total is 100 points. A problem

which has the word prove means that you must provide a general proof. Pleaseturnoff your

C E L L

P HON E before the exam. Please write your

F U L L

N AM E on the front of your

bluebook.

1. (2 points

each:

10 points total) Give complete definitions of the following terms. Onlyminor

mistakes wi l lbe acceptable tostillget 2 points, otherwise 0 points.

a. theSpano f aset ofvectorsS = { v l 5 . . . , vk}

b. alinearly independentset of

vectors

5 =

{ v i , . . . ,

v

k

}

c asubspace Wof K .

d

abasisfor a

subspace

Wof

R .

e. thedimensionof a

subspace W of

R .

2

(2 points

each:

10 points total) Indicate i f the statement is True or False, and give a brief

explanationwhy.

Answerandreasoning must both be correct to get 2 points, otherwise, 0 points.

a.

f

5 = { v i , V 2 , V 3}is a set ofthree non-zero vectors

from

K 3 , thenSpan S) isallof K 3 .

b.

Suppose

that a

subspace W

has dimension 6. Then a set of 10 vectors

from W

is

automatically

linearlydependent.

c

If

S = { v i , v 2 , V 3 , V 4 , v 5 }

is a

linearly

dependentset, then two of

these

fivevectors must be

paralleltoeachother.

d

I fWisa 5-dimensional

subspaceo f

O S 8 thenW

L

is also 5-dimensional.

e.

f

.4 is a 6x10matrix,and its rank is 4, then its

nullity

is 2.

3

PROOFS: Give a complete andconvincing

proof

for

each

of the following. You may

assume

the

11Axiomsin Chapter Zero as wel las any Theorems proven in theTEXT (NOT inthe Exercises),

as

wel l

as the closure properties for integers anddefinitionsfor even and odd numbers:

a. (6 points) Prove that for

a ll

a,

be i f

a

b

is even, then either

a

is even

or b

is even. Hint:

start by stating what strategy youwi l lbe using.

b. (4 points) Prove that for

a ll

v

G

R , 0

v = 0.

c (4 points) Prove that fora llk e OS k 0 = 0.

d

(10 points) Prove that fora llke K, and fora llv e 1 :

kv= 0 if and only if = 0orv = 0.

Y ou are free to use (b) and(c),even i fy ou did not do them,inyour

proof

for (d ).

T H E R E

IS A

B A C K P AG E


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6. Let A =

3 4 . - 2

-5 -7 1

-3 0

1 6

5 -2

C =

z

-2 -3 4

3

7 - 1

and

2 -5

-3 7

Compute the following, i fpossible:

a . 3 points)5A- 3C

b. 3 points) AS

C .

3 points)

BA

d. 5 points)p D), where X*) = 4x3 - 5x2 -

Tx

+ 3

7. 4 points ) Show theeffecton the basic box produced by

letter R inside this box.

-2 3

3

5

Show what happens to the

8. 10 points) Suppose that T: I R2 -> OS3, and you are told that r - 3 , 5 = -2,6,4) and

r 5, - 8 = -2,1,-5). Find7I


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6/10

5. 11points) LetV= I P

3

the

set

of polynomials ofdegree

at

most3,

and:

W =

{p(x)\p(3) =2p(-l)

and

/ r ( l )

= 3p(2)}.

a.

Show that

Wis

asubspaceof

V.

b. Find

a

basis

for

Wcontaining polynomials w i th only

integer coefficients, andstatethe

dimension of

W.

6.

10

points) Decide whether

or not the set of

functions S

{cos

2

(x),cos

2x),7r}

is

linearly

dependentor

independent.

Defend your decision.

7. 12points) LetS

=

{1

- x , l +x>mdB

= {1,1

- x , + x

2

} .

a. Explain whyB

is a

basisfor I P 1andB

1is

abasisforI P 2 .

b. SupposethatT :IP

1

-

I P

2

is

given by:

4 -10

I T W =

-10 25

6

-15

Compute7 5x

-

4). Hint/warning:BandB

1

are NOT thestandard

bases

for

I P

1

and

P

2

.

c.

Find a

basis

for ker(F) and for

range(T).

d. IsTone-to-one?IsTonto?

e. State

the Dimension Theorem in general, and

verify

it for

T.

8. 12points) Find

a

function/ x)which is

a

solution to the differential equation:

4/ x) -7/ x) =5e7xcos3x)- 9e x sin3x).

9. 12points) Let

T

:

P

3 -+

I P

2

be

given by:

r p x))

=

3y x)

+

p 0) x2-5).

a. Warm-up: Compute

7 7 -

2x + 4x3).

b. Explain in general whyT p x))

e P2

foranyp(x)

e P3.

c. Show thatTisindeed

a

linear transformation.

d. LetB

=

{ l , x , x

2

, x

3

}

and

B

=

{ l , x , x

2

} ,

the

standard

bases for

Find [T ] BiB .

and P2 , respectively.

e.

Check that your

answer

to(d)iscorrect by re-computing(a)using your

answer

to(d).

f.

F i l lin the blank:Tcannotbe


7/10

Math10

Fourth

Examination

Instructions:

Show complete solutions to the following problems. An answer without a solution

will

get little or no credit. Do not write your answers on thissheetKeep it after the

examination.T h e total is 100 points.

1. (5 points each)

I n

a shortparagraph,discuss:

a . the terms: eigenvalues, eigenvectors, and

eigenspaces

for a square

matrix

A Thisparagraph

should specifically address the role of the zero vector.

b. the concept of diagonalizabihty, mentioning the definition and relevant theorem/s.

2. Suppose matA andB are any nun matrices. Prove the following statements:

G

a .

(% points)

Th e characteristic

polynomial of

A

is the same as that of A T

b . ( points) I f vis an eigenvector for an invertible

matrix

Awith corresponding eigenvalueX,

then v is also an eigenvector for

A

k

for all positive integers

k.

What is the corresponding

eigenvalue?

c . (10 points) SupposeA is a 7 x 7 matrix,

{vi ,v2 }

is a basis for

Eig(/1,3),

{ v 3 , V 4 , v 5 } is a

basis for

Eig(4,7),

and {v$,v7} is a basis for

Eig(^4,-2).

Prove that the set

{v i ,V2 ,V3 ,V4 ,V5,V6 ,V7}

islinearlyindependent.

3. (10 points)L eta = (5,7,10,6,2,8,1,4,9,3).

a .

Find

the number of inversions of

a.

List al lof them.

b.

Finda

1 How many inversions should it

have?

Do not list them

4. (5 points) Suppose that we have thematrix:

0 0 0 7

0 -3 0 0

0 0 4 0

-2 0 0 0

Applythe determinant formula directly in order to compute det(4). Thereshould be only one

term:find thisterm,the permutation that produced this term,and whether the permutation is even

or

odd, as part of your solution to find det(4). DO NOT use row operations or cofactor

expansion.

T H E R E

ISAB A C K P A G E


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5. 10 points) Use any method or combination of

methods

that you wish in order tocomputethe

following determinant:

3 2 - 4 2

-5 0 3 -8

1 - 3 2 3

-4 5 - 2 0

6. 10 points) Suppose you were told that the determinant of

thematrixA

given

below

on the

left

is

-15.

Find

the determinant of the

matrix

B

on

the right:

d\2 a3

b\

2

bi

a c2 c

B =

3ax

+5ci

3a2 +5c

2

3a3 +5e

3

lc\-lb\2-2b2

7 c

3

- 26

3

7. 10 points) Decide whether or not the following matrix is diagonalizable, and

give

a brief

explanation as to how you obtained your decision. Note/Hint: You are not being asked to actually

diagonalize thematrix.Youareonly being asked to determine whether or not it is diagonalizable.

4 0 0 0

3

-7

0 0

-6 22

4

0

18

-66

-33 -7

8.

20 points)

Consider

the

matrix:

-70 -9 18

A

=

72

11

-18

-216

-27 56

a. Diagonalize

A You

must provide the invertible matrix

C, and

the diagonal

matrixD,

as well

as

C 1 . Thecorrect characteristic

polynomial is:

p X)

=

A

3

+

3X2

-

24A

+ 28.

Youarestill

expected

to show ah thestepsto getp[X) correctly. You are given this answer

so

that you

willnot be wasting

tim

on the wrong polynomial.

b. Use a) tocompute

A6.


9/10

Math

10

Fifth

Exam

Instructions:Show complete solutions to the following problems on your bluebooks. An answer

without a solution, even ifcorrect,willget little or no credit. The total is 100 points.

T U R N O F F

Y O U R

C E L L

P H O N E

1. 55 points total)Considerthebilinearform on I P2 :

a . 6 points) Show that this bilinear form is symmetric, additive andhomogeneous.

b.

4 points) Prove that this bilinear form hasthepositivityproperty.

C . 4 points)

Explain

why this bilinear form isnotan inner product on P

3

. You may want to

provide a counterexample ofsomekind.

d.

6 points)F i n dthe exact cosine and the approximate value in

radians)

of the angle6between

3

+2x - 4x2 and2 + 5xwith respect to this inner product.

10 points) LetW - Span {3 +2x-4x

2

}). F inda basis for W

L

consisting of polynomials

withintegercoefficients withoutusing the

Gram-Schmidt

Algorithm.Hint:you can use part

of

your

work from d).

)

f. 10 points) F i n d an orthonormal basis for IP2using the set {x2,x, 1} as input to the

Gram-SchmidtAlgorithm.

^ g . 2 points) Let U = Span {x2}) F i n d an orthonormal basis for UL without any further

computations using your answer in f).

~4

h. 8 points)

L e tp x) = 3x2 -7x + 9.

F i n d

p x)

B

, where

B

is the orthonormal basis mat you

found in f).

^4yi. 5 points)

F i n d

the orthogonal decomposition ofp x) with respect to U, where U is the

subspace in g) andp x) is the polynomial in h).

T H E R E IS A

B A C K P A G E

p x) q x) ) = p -l )q{-\) +p{\)q{\) +p 2)q 2).


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2 . (10 points) Suppose that V is a vector space and T :V~ R is an

one-to-one

linear

transformation.

Prove that the bilinear form:

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