[PDF] [PDF] Chapter 7 Linear Transformations

For a linear transformation, the number of elements in the set K(w) {v : T(v) = w} A non-square matrix A does not have “inverse” (but may have left-inverse or



Previous PDF Next PDF





[PDF] Generalized Inverse of Linear Transformations: A Geometric - CORE

A generalized inverse of a linear transformation A: V -+ W, where 7v and Y are arbitrary finite dimensional vector spaces, is defined using only geometrical



Generalized Inverse of Linear Transformations - ScienceDirectcom

A generalized inverse of a linear transformation A: V -+ W, where 7v and Y are arbitrary finite dimensional vector spaces, is defined using only geometrical



[PDF] Which Linear Transformations are Invertible

We have mentioned taking inverses of linear transformations But when can we do this? Theorem A linear transformation is invertible if and only if it is injective 



[PDF] (Lecture 28 Compositions and Inverse Transformations [???e????])

2 jan 2012 · prove that V is isomorphic to Rn we must find a linear transformation T:V→Rn that is Inverse Linear Transformations ▫ A matrix operator T A



[PDF] Chapter 7 Linear Transformations

For a linear transformation, the number of elements in the set K(w) {v : T(v) = w} A non-square matrix A does not have “inverse” (but may have left-inverse or



Linear Transformations - Penn Math

23 juil 2013 · mapping T : V → W is called a linear transformation from V to W if it inverse transformation if and only if A is invertible and, if so, T−1 is the 



[PDF] Chapter 4 LINEAR TRANSFORMATIONS AND THEIR - TAMU Math

The central objective of linear algebra is the analysis of linear functions defined on a finite so that we could call Mu a left inverse of MA However, MA Mu ' 3



[PDF] Linear Transformations

Def: A linear transformation is a function T : Rn → Rm which satisfies: (1) T(x + y) Question: If inverse functions “undo” our original functions, can they help

[PDF] inverse of matrix product

[PDF] inverse relationship graph

[PDF] inverse relationship science

[PDF] inverseur de source courant continu

[PDF] inverter layout

[PDF] invertible linear transformation

[PDF] invest in 7 eleven

[PDF] investigatory project in physics for class 12 cbse pdf

[PDF] investing in hilton hotels

[PDF] investment grade rating

[PDF] investor pitch presentation example

[PDF] investor presentation (pdf)

[PDF] investor presentation ppt template

[PDF] invité politique dimanche france inter

[PDF] invité politique matinale france inter

Chapter 7

Linear TransformationsPo-Ning Chen, Professor

Department of Electrical Engineering

National Chiao Tung University

Hsin Chu, Taiwan 30050, R.O.C.

7.1 The idea of a linear transformation

7-1

•AtransformationTis simply a mapping fromVtoW.

•It is sometimes denoted asT:V?→W.

•A transformation is linear if

1.T(v 1 W T(v 2 )=T(v 1 V v 2

2.T(c·

V v)=c· W T(v) where "+ W V "and"· W V " denote some general "additions" and "mul- tiplications" defined overWandV, respectively.

VandWare usually vector spacesVandW.

For simplicity, we will drop the subscripts in "+" and "." (in case there is no ambiguity in these operations).

7.1 The idea of a linear transformation

7-2

Important notes on linear transformation

•A line segment will be transformed to a line segment. T(a 1 v 1 +(1-a 1 )v 2 )=a 1 T(v 1 )+(1-a 1 )T(v 2 )=a 1 w 1 +(1-a 1 )w 2 •Hence, a triangle will be transformed into a triangle.

•0inVwill be transformed to0inW.

T(0)=T(0·v)=0·T(v)=0.

Note again that the0inVand the0inWmay be different. For example, T ?0 0 0? =?0 0?

7.1 Kernel

7-3 Definition (Kernel):Thekernelof a transformationTis the set of allvsuch that

T(v)=0.

•The concept of "kernel" becomes more evidently important when the transfor- mationTis linear. •For a linear transformation, the number of elements in the set

K(w)?{v:T(v)=w}

is a constant, independent ofw.

Proof:

-Suppose distinctv 1 ,v 2 ,v 3 ,...,v k satisfyingT(v i k,wherek=|K(w)|is the size of the setK(w). Then, either|K(˜w)|≥k or|K(˜w)|= 0 because for a given˜vsatisfyingT(˜v)=˜w,wehave

T(˜v+v

i -v 1 )=T(˜v)+T(v i )-T(v 1 -Since we can interchange the role ofwand˜w, we conclude that|K(w)|= |K(˜w)|if they are positive.?

7.1 Kernel

7-4 •Note that sinceT(0)=0for a linear transformation,K(0) cannot be empty. So|V| |K(0)| will give the number of elementswinWsuch thatT(v)=wfor somev. Definition (Range):Therangeof a transformationTis the set of allwsuch that

T(v)=wfor somev.

I.e., {w?W:?vsuch thatT(v)=w}.

Important fact about linear transformation

•A linear transformation from a vector spaceVto a vector spaceWcan always be represented as Av=w by properly selecting the matrixA.

•So,?

Kernel=Null spaceofA

Range=Column spaceofA

7.1 Problem discussion

7-5 (Problem 16, Section 7.1) SupposeTtransposes every matrixM. Try to find a matrixAwhich givesAM=M T for everyM. Show that no matrixAwill do it. To professors:Is this a linear transformation that doesn"t come from a matrix. Thinking over Problem 16:Define a transformation that maps a matrixM

2×2

to its transposeM T . Is this a linear transformation?

Solution.

•?T(M

1 +M 2 )=(M 1 +M 2 T =M T1 +M T2 =T(M 1 )+T(M 2

T(c·M)=(c·M)

T =cM T =c·T(M) hence, it is a linear transformation.?

7.1 Problem discussion

7-6

•There does not exist any matrixA

2×2

satisfyingAM=M T

•But there does exist a matrixA

4×4

satisfying A? ??m 1,1 m 1,2 m 2,1 m 2,2 ??m 1,1 m 2,1 m 1,2 m 2,2 So a linear transformation can always be represented asAv

4×1

=w

4×1

(since the dimension ofMisfour).

7.2 The matrix of a linear transformation

7-7

For a linear transformation

T:V?→W,

how to find its equivalent matrix representation A m×n v n×1 =w m×1

Answer:

•Denote the standard basis for vector spaceVby

e 1 ??1 0... 0? ??,e 2 ??0 1... 0? ??,...,e n ??0 0... 1?

•Then,T(e

i )=Ae i gives ?T(e 1 )T(e 2 )···T(e n )?=A?e 1 e 2

···e

n ?=A.

7.2 The matrix of a linear transformation

7-8 Example.v(x) = a polynomial ofxof order 3, andT(v)=∂v(x) ∂x.

•The bases forv(x)are1,x,x

2 ,x 3 . Or in vector forms,? ??1 0 0 0? ??0 1 0 0? ??0 0 1 0? ??0 0 0 1?

•So,A=?T(1)T(x)T(x

2 )T(x 3 )?=?012x3x 2

Or in matrix form,A=?

?0100 0020 0003?

7.2 The matrix of a linear transformation

7-9

•Hence, ifv(x)=1+2x+x

3 ??1 2 0 1? ??,then ∂v(x) ∂x=A? ??1 2 0 1? ?2 0 3? =2+3x 2

7.2 The matrix of a linear transformation

7-10

For a linear transformation

T:V?→W,

how to find its equivalent matrix representation A m×n v n×1 =w m×1 (by the bases other thane 1 ,e 2 ,...,e n

Answer:

•Denote a basis for vector spaceVbyv

1 ,v 2 ,...,v n

•Denote a basis for vector spaceWbyw

1 ,w 2 ,...,w m

•Suppose that

T(v i )=b 1,i w 1 +b 2,i w 2 +···+b m,i w m

Then,T(v

i )=Av i gives A ?v 1 v 2

···v

n ?=?T(v 1 )T(v 2 )···T(v n )?=?w 1

···w

m ???b 1,1 b 1,2

···b

1,m b 2,1 b 2,2

···b

2,m b m,1 b m,2

···b

m,m

7.2 The matrix of a linear transformation

7-11

Hence,

A=?T(v

1 )T(v 2 )···T(v n )??v 1 v 2

···v

n -1 =?w 1

···w

m ??b 1,1 b 1,2

···b

1,m b 2,1 b 2,2

···b

2,m b m,1 b m,2

···b

m,m ???v 1 v 2

···v

n -1 Example (Example 6 in the textbook):Tprojects a vector in? 2 onto the line passing via (0,0) and (1,1). Find the projection matrixA.

Solution 1:

•A=?

T??1 0?? T??0 1??? 121
21
21
2

7.2 The matrix of a linear transformation

7-12

Solution 2:

•Choosev

1 1 21
2 andv 2 1 21
2

•ThenT(v

quotesdbs_dbs20.pdfusesText_26