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

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Matrices and Linear Systems

Advanced Engineering Mathematics by

Erwin Kreyszig

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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Pages 274-275

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Advanced Engineering Mathematics by Erwin Kreyszig

3/13/21

formd echelon row reducethe

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186x4x

systems.nt inconsiste are that theyShow 2.

of formsechelon reduced row Find 1.

matrix.echolon reduced row a is 20100

15031:

Example

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Continued

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Vector Spaces

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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

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avauvua

buauab

buauuba

wvuwvu

VauvuRaVvu

scalar)identity is (1 1.8

)().(7

inverse) additive uique (a 0)( .5

element) zero unique (a 0 .4

)()( .3

. , then , and , If.1

Examples for vector spaces

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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

{0}V .1

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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

ucucuc

DependenceLinear

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

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toequivalent row is

Example

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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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Example

0101,3021

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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Continued

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Continued

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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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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

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

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Pages 331-332c

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

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