Matrices and Linear Systems - DEUkisi.deu.edu.tr/melda.duman/ch2+ch7-matrix,inner product.pdf ·...

Matrices and Linear Systems

Advanced Engineering Mathematics by

Erwin Kreyszig

Copyright 2007 John Wiley & Sons, Inc.

Roughly speaking, matrix is a rectangle array

We shall discuss existence and uniqueness of

solution for a system of linear equation.

The method of Gauss ellimination will be given to

solve the system .


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Page 274 (1)


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Page 274 (2)


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Pages 274-275


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Page 275 (1)


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Page 275 (2)


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Page 276 (1)


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Page 279a

Continued


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Page 279b


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Page 280 (1)


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Page 282


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Page 283 (1)


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Page 283 (3)


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Page 288 (1)


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Page 288 (2a)

Continued


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Page 288 (2b)


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Page 291a

Continued


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Continued


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Page 291c


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Page 292 (1)


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Page 292 (3)

Advanced Engineering Mathematics by Erwin Kreyszig

Copyright 2007 John Wiley & Sons, Inc. All rights reserved.

Page 294a

000

110

101

000

110

3/13/21

formd echelon row reducethe


Erwin Kreyszig


186x4x

632x

34xx

03xx

132x

systems.nt inconsiste are that theyShow 2.

102

110

312

,

1000

0000

1141

of formsechelon reduced row Find 1.

matrix.echolon reduced row a is 20100

15031:

21

21

21

21

21

xx

Example


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Page 294b

Continued


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Page 294c

Vector Spaces


Erwin Kreyszig


A quantity such as work, area or energy which is

described in terms of magnitude alone is called a scalar.

A quantity which has both magnitude and direction for

its describtion is called a vector.

A vector is an element of vector space.

Definiton: A vector space V in R is the set satisfying


Erwin Kreyszig


avauvua

uu

buauab

buauuba

uu

uu

wvuwvu

uvvu

VauvuRaVvu

)(.9

scalar)identity is (1 1.8

)().(7

)( .6

inverse) additive uique (a 0)( .5

element) zero unique (a 0 .4

)()( .3

.2

. , then , and , If.1

Examples for vector spaces


Erwin Kreyszig


func.) abledifferentily (continous b][a,C 6.

b])[a,on space -(function b]F[a, 5.

s)polynomial of (space [x]P .4

matrices) of (space R .3

.2

{0}V .1

n

n

mxn

nR


Erwin Kreyszig


dependent.linearly are then theyzero is any if

e,t.Otherwisindependenlinealy are ,...,

then , allfor solution alonly trivi has

0...equation theIf

,...3,2,1,, where

,...

vectorsnonzero ofn combinatiolinear a Define

i

21

i

2211

2211

c

uuu

c

ucucuc

iRcVu

ucucuc

n

nn

ii

nn

DependenceLinear


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Page 297 (2)


Erwin Kreyszig


Page 298 (2)

Rank of A is 2 because the first two rows are linearly independent.


Erwin Kreyszig


3

.

100

010

001

toequivalent row is

102

110

312

rankA

A

Example


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Page 298 (3)


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Page 299

Dimension of a vector space V


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SpanS= All linear combinations of vectors of the subset

S of V .

A basis for V is a linearly independent subset S of V

which spans the space V.

That is, SpanS= V where S is lin. İndep.

dimV= The number of vectors in any basis for V.

V is finite-dimensional if V has a basis consisting of a

finite number of vectors.

Note: (6) is known as dimension theorem


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Pages 302-303a

Continued


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Pages 302-303b


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Page 304


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Page 305 (1)


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Page 305 (3)


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Th

Tcxx

xxxx

xxxx

xxxx

Example

0101,3021

22

30

12323

0

4321

4321

4321

Determinant


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Determinant is a function form square

matrices to scalars.

Our efficient computational procedure will be cofactor

expansion.


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Page 306a

Continued


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Continued


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Continued


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Page 309 (1)


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Page 309 (2a)

Continued


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Page 309 (2b)


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Page 310


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Page 311 (1)


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Page 312a

Continued


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Page 316


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Pages 317-318a

Continued


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Pages 317-318b

Continued


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Pages 317-318c


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Page 318


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Page 319a

Continued


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Examples


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Linear Transformations


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Examples: Zero transform, identity operator, scalar-multiple

operator,reflection , projection , rotation, differential

transform, integral transform.

Representiation Matrix


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Example


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Example:

Find the representiation matrix of


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Range and Null (Kernel) spaces


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V.nullityFrankFTheorem

rankF(RangeF)

nullityF(NullF)

WVuuFvvRangeF

VuuFuNullF

nsform.linear traWVF

dim :

dim

dim

vectors.images all includes

}),(:{

}.,0)( :{

a be :Let


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Pages 331-332a

Continued


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Pages 331-332b

Continued


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Pages 331-332c

Continued


Erwin Kreyszig


Pages 331-332c

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