linear algebra lecture2

8/10/2019 Linear Algebra Lecture2

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Synopsys University Courseware

2009 Synopsys, Inc.Lecture - 2Developed By: Vazgen Melikyan

Linear Algebra


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Linear Systems: Introduction1 lecture

Defined Linear Spaces on the Given Field3 lectures

Linear Subspaces1 lecture Linear Transformation

1 lecture

Matrices and Determinants2 lectures

Linear Systems : Continuation2 lectures

Linear Transformation. Continuation2 lectures

Content (1/2)


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a

a

vector

3

2

1

x

x

x

a

3

2

1

x

x

x

3

2

1

y

y

y

a

a

x1

x2

x3

b

ba

33

22

11

y x y x y x

ba

a

32

1

x x x

Linear operations: Example (1)


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3

2

1

3

2

1

2

3

2

1

2

23212

)(2

)(

)(

,

y

y

y

xQ

x

x

x

x

x

x

x

x

j x x x x x x x

33

22112

33221122 y x y x y x

x y x x y x y x xQ x

3212

3112 x x x

x x x x x x

fixed real numbers

Linear operations: Example (2)


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Linear Space (Vector Space): Definition

A vector space (over ) consists of a set along with two operations ,,+and ,, so that 1. If , their vector sum , and

there is zero vector so that each has an additive inverse so that

2. If scalars) and then each scalar multiple is in , and 1 =

3. distributive laws (connection of ,,+ and ,,)

L y x , L y x x y y x

)( z y x z y x L0 L x x x 0

L x L y 0 y x L y x ,

x sr xrs x

xr

x s xr x sr

x

yr xr y xr

R L

R sr , L


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1.

nk R x L k

n x

x

x

,1,:2

1

with operations

R

x

x

x

y x

y x

y x

n y

y

y

n x

x

x

nnn

,;

1

,

1 111

2. Ra xa xaa x P L k nnn ;)( 11101 with natural operationsaddition and scalarmultiplication

1n D L

n R L

Examples of Vector Spaces


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1. Definition: A subset of a vector space is linearly independent if none of its

elements is a linear combination of the others. Otherwise it is linearly

dependent.

2. Definition: A subset S of a vector space is linearly independent if and only iffor any distinct the only linear relationship among

those vectors

is trivial one:

S x x m ,,1

mk R x xk

mm ,1,,011

021 m

Linear Independence


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1. The set is linearly independent in

Because

2. In where

linearly dependent

x x 1,1 2,1,0,22 210

k Ra xa xaa D k

00

0

20000)()()1()1(

2121

21

212121

x x x x x

4

4

10

,

2

2

5

,

1

2

3

321 x x x

321321 ,,

0

0

0

0)1(20 x x x x x x

3 R

Examples


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Definition: If there exists linearly independent vectors in the linearspace , and arbitrary n+1 vectors in the linear space , are linearlydependent, then the dimension of is or that is -dimensional space.Example: The dimension of the set of first order polynomials is 2.1. 1 and x are linearly independent:

2. for arbitrary the

and

001 2121 x

01011

,,1

110

1101

x P xaa

P x xaa x P

nee ,,1 L L

L n L n

is linearly dependent

Dimension of Vector Space


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Definition: A basis for a vector space is a sequence of vectorsthat form a set that is linearly independent and that spans thespace.Example: This is a basis for

1. It is linearly independent

2. And it spans

for arbitrary .

1

1,

4

22 R

004

02

0

0

4

2

4

221

21

2121

2

2,24

211

42

21

22

2121

R y

x

y xc x yc ycc xcc

y xcc

2 R

Basis


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For any

k k

n

e

ee

0

1

0

,,...,1

where -th row is the standard(or natural) basis in

n R

n R

Basis


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Assume is n-dimensional vector space, and -basis in . Then uniquely represents in the form.

Here th coordinate of the vector in basis.That is

n f f ,...,1 La

R x f xa k n

k k k

,1

a n f f ,...,1

n x

xa

1

k xk

L L

Coordinates


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Definition: An isomorphism between vector spaces L and Mis a map , that1. is a correspondence one-to-one and onto

2. preserves linear structure:if , then

xrf xr f Rr and L x

y f x f y x f

M L f :

L y x ,

Isomorphism


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1. Arbitrary n-dimensional linear space L is isomorphic to

2. Vector spaces are isomorphic if and only if they havethe same dimension.

nk Rk xn x

xn

R ,1,:

1

Properties of Isomorphic spaces

linear algebra lecture2

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