Chapter 4 Linear Transformations. Outlines Definition and Examples Matrix Representation of linear transformation Similarity.

Chapter 4Linear Transformations

Outlines

Definition and Examples Matrix Representation of linear transformation Similarity

Linear transformations are able to describes. Translation, rotation & reflection Solvability of Dx &

Ax b

Definition: A mapping L from a vector space V

into a vector space W is said to be a

linear transformation (or a linear

operator) if

Remark: L is linear

1 2 1 2

1 2

( ) ( ) ( )

for all , V & , F

L v v L v L v

v v

1 2 1 21 2

( ) ( ) ( ) for all v , , V and F

( ) ( )

L v v L v L vv v

L v L v

Example 1:

Remark: In general, if , the linear transformation

can be thought of as a stretching ( ) or shrinking ( ) by a factor of

( ) 3 is linearL x x

( ) 3( ) (3 ) (3 )

= ( ) ( )

L x y x y x y

L x L y

0

( )L x x

0 1 1

Example 2:

1 1

1 1 1

1 1 1 1

1

( ) is linear

since ( ) ( )

( ) ( )

( ) ( )

In fact, L can be thought of as a projection onto the -axis.

L x x e

L x y x y e

x e y e

L x L y

x

Example 3:

1 2

1 1

2 2

1 1

2 2

1

( ) ( , ) is linear

since ( )( )

( ) ( )

In fact, L has the effect of reflecting vectors about the x

TL x x x

x yL x y

x y

x yL x L y

x y

axis

Example 4:

2 1

2 2

1 1

2 2

1 1

2

( ) ( , ) is linear

( )since ( )

( ) ( )

In fact, has the effect of rotating each vector in R by 90

in the

TL x x x

x yL x y

x y

x yL x L y

x y

L

counterclockwise direction

Example 8:

1 2 1

If is any vector space, then the identity operator is

defined by

( )

for all . Clearly, is a linear transformation that maps

into itself.

( )

V I

I v v

v V I

V

I v v v

2 1 2( ) ( )v I v I v

1Let be the mapping from [ , ] to defined by

( ) ( )

If and are any vectors in [ , ], then

( ) ( )( )

( )

b

a

b

a

L C a b R

L f f x dx

f g C a b

L f g f g x dx

f x d

( )

( ) ( )

Therefore, is a linear transfromation.

b b

a ax g x dx

L f L g

L

Example 9:

Example 10:

1Let be the operator mapping [ , ] into [ , ]

defined by

( ) (the derivative of )

is a linear transformation, since

( ) ( ) ( )

D C a b C a b

D f f f

D

D f g f g D f D g

Lemma:

1 1

Let : be a linear transformation.

Then

(i) (0 ) 0

(ii) ( ) ( )

(iii) ( ) ( )

Pf: (i) Let 0, (0 ) (0 0 ) 0

(ii) By mathematical induction.

(iii) 0 (0

V W

n n

i i i ii i

V V W

W

L V W

L

L v L v

L v L v

L L

L

) ( ( )) ( ) ( )

( ) ( )V L v v L v L v

L v L v

Def:

Let : be a linear transformation and let be

a subspace of . Then

(1) ( ) { | ( ) 0 } is called of .

(2) ( ) { | ( ) }

is called the of .

(3) The image

W

L V W S

V

Ker L v V L v kernel L

L S w W w L v for some v S

image S

of the entire vector space, ( ),

is the of

L V

range L

Theorem 4.1.1:

Let : be a linear transformation, and be a

subspace of .

Then (i) ( )

(ii) ( )

Pf: trivial

L V W S

V

Ker L is a subspace of V

L S is a subspace of W

Example 11:

2 2

1 21 1 1

11 2

2

Let be the linear transformation form into defined by

( ) ( ) { }0

0 0 ( ) =span

0

L

xL x x e L span e

xL x Ker L e

x

Example 12:3 2

1 2 2 3

31 3

1 2

2 3

Let : be the linear transformation defined by

( ) ( , )

and let be the subspace of spanned by and .

00 ( )

00

T

L

L x x x x x

S e e

x xL x

x x

1 3

2 3 2

1

1 , .

1

1

( ) 1

1

0 and , 0 : ,

( ) ( ) .

x a a

Ker L span

a aa

L S span e e a bb

b b

L S L

Example 13:

2

3 3

1

3 2

: P P

'

2

0 0

ker( ) P {1}

(P ) P

D

p p

a bx cx b cx

D(p) b c p a

D span

D

Theorem:

(one-to-one)

Let : be a linear transformation.

Then L is an injection ( ) {0 }.V

L V W

Ker L

1 2 1 2

1 2 1 2

pf: L is one-to-one

( ) ( )

( ) 0 0

ker( ) {0 }

W V

V

L x L x implies that x x

L x x implies that x x

L

§4.2 Matrix Representations of Linear Transformations

Theorem4.2.1:

if is a linear operator mapping into , there is an

matrix such that

( )

for each . In fact, the th column vector of is given by

(

n m

n

j

L

m n A

L x Ax

x j A

a L

) 1, 2,...je j n

1 2Remark: [ ( ) ( ) ... ( )] is called as the standard

matrix representation of L.nA L e L e L e

Proof:

1 2

1 2

1 1 2 2

For 1,..., , define

( , ,..., )

Let

( ... )

If

...

then

Tj j j mj j

ij n

nn n

j n

a a a a L e

A a a a a

x x e x e x e

L x

����

1 1 2 2

1 1 2 2

1

1 2

( ) ( ) ... ( )

= ...

=( ... )

n n

n n

n

n

x L e x L e x L e

x a x a x a

x

a a a

x

���

Example:

3 2

11 2

22 3

3

3

1 2 3

11 2

22 3

3

Let :

Find a matrix A ( ) .

1 1 0, ,&

0 1 1

1 1 0

0 1 1

1 1 0and (

0 1 1

L

xx x

xx x

x

L x Ax x

L e L e L e

A

xx x

Ax x Lx x

x

3), x x

Solution:

Example: 2 2Determine a linear mapping from to

which rotates each vector by angle in the

counterclockwise direction. Find the standard

matrix representation of .

L

L

1 2

cos cos( )Let , then

sin sin( )

cos( )cos sin2 , sin cos

sin( )2

cos -sin cos cos( )

sin cos sin sin(

r rL

r r

L e L e

r rAx

r r

( )

)L x

Solution:

Figure 4.2.1:

(0,1)

(cos ,sin )

(-sin ,cos )

(1,0)

Ax

x

Question: Is it possible to find a similar representation for a linear

operator from to , where and are general vector

spaces with dim and dim ?

V W V W

V n W m

1 2 1 2

1 1 2 2

i.e. Let [ , ,..., ] and [ , ,..., ] be two

ordered bases for and , respectively.

Let :

..

n mE v v v F w w w

V W

L V W

v x v x v

1 1 2 2

1 1

2 2

. ( ) ...

Hence [ ] = = and [ ( )] = =

n n m m

E F

n m

x v L v y w y w y w

x y

x yv x L v y

x y

1 1 2 2

Does there exist an matrix representing the operator

such that

( ) ... ?

m m

L

m n A L

y Ax L v y w y w y w

v V

( )

[ ] [ ( )]An mE F

L v W

x v R y Ax L v R

Theorem4.2.2:

1 2 1 2If [ , ,..., ] and [ , ,..., ] are ordered bases for vector

spaces and , respectively, then corresponding to each linear

transformation : there is an matrix such that

n mE v v v F w w w

V W

L V W m n A

[ ( )] [ ] for each

is the matrix representating relative to the ordered bases and .

In fact, [ ( )] 1, 2,...,

F E

j j F

L v A v v V

A L E F

a L v j n

Denoted by F

EA L

1 1 2 2

1

2

1 2

1 1 2 2

Proof): Let ( ) ... 1

[ ( )]

Let ( ) [ ]

If ... , then

j j j mj m

j

j

j j F

mj

ij n

n n

L v a w a w a w j n

a

aa L v

a

A a a a a

v x v x v x v

1 1 1 1 1 1

1 11

2

1

( ) ( ) ( ) ( ) ( )

[ ( )] [ ]

n n n m m n

j j j j j ij i ij j ij j j i i j

n

j jj

F E

n

mj j nj

L v L x v x L v x a w a x w

a x x

xL v y A Ax A v

a x x

Example 3:

3 2

1

2 1 1 2 3 2 1 2

3

1 2 3 1 2

Let :

1 1 ( ) where and

1 1

Find the matrix , where [ , , ] and [ , ]F

E

L

x

x x x b x x b b b

x

A L E e e e F b b

Solution:

1 1 2

2 1 2

3 1 2

1

2 3

( ) 1 0

1 0 0 ( ) 0 1

0 1 1

( ) 0 1

Check: ( ) [ ]EF

L e b b

L e b b A

L e b b

xL x Ax A x

x x

Example 4:

1 1 2

2 1 2

1 1( ) 1 0

0 2 ( ) 1 2

Check: ( ) [ ]2 EF

L b b bA

L b b b

L x A A x

2 2

1 2 1 2

1 2

Let :

( ) 2 .

Find the matrix , where [ , ] is defined

in example 3.

F

F

L

x b b b b

A L F b b

Solution:

Example 5:

3 2

2

2

2

Let D :

2

Find where , ,1 and ,1

( ) 2 0 1

2 0 0 ( ) 0 1 1

0 1 0

F

E

P P

p c bx ax p b ax

A D E x x F x

D x x

D x x A

(1) 0 0 1

2 check : ( )

F E

D x

aa

D p A b A pb

c

Solution:

Theorem 4.2.3

1 1Let ,..., and ,..., be ordered bases

for and , respectively. If : is a linear

transformation and A is the matrix representing with

respect to and , then

n m

n m n m

E u u F b b

L

L

E F

1

1 2

( ) for 1,...,

where ( ... )

j j

m

a B L u j n

B b b b

Proof :

1 1 2 2

1

2

1 2

( ) j 1,2,...,n

( ) ... j 1,2,...,n

...

F

j jE F

j j j mj m

j

j

m

mj

A L a L u

L u a b a b a b

a

ab b b

a

1

11 2

( ) j 1,2,...,n

( ) ( ) ( )

j

j j

n

Ba

a B L u

A B L u L u L u

Cor. 4.2.4:

1 2

1 1 1 11 2 1 2

1 2

( | ( ) ( )... ( )) is row equivalent to

( | ( ) ( )... ( )) ( | ( ) ( ) ... ( ))

( | ... )

n

n n

n

B L u L u L u

B B L u L u L u I B L u B L u B L u

I a a a

( | )I A

Proof: 1 2

The reduced row echelon form of

( | ( ) ( )... ( ) is ( | )).nB L u L u L u I A

Example 6 :

2 3

11

1 22

1 2

1 2

1 2 3

Let :

1 3Let , ,

2 1

1 1 1

, , 0 , 1 , 1

0 0 1

Find F

E

L

xx

x x xx

x x

E u u

F b b b

A L

Solution(Method I):

1 2

1 2

2 1

( ) 3 and ( ) 4

1 2

1 1 1 2 1 1 0 0 -1 -3

( | ( ) ( )) 0 1 1 3 4 0 1 0 4 2

0 0 1 -1 2 0 0 1 -1 2

1 3

4 2

1 2

L u L u

B L u L u

A

1 1 2 3

2 1 2 3

2

( ) 3 4

1

1

( ) 4 3 2 2

2

1 3

4 2

1 2

L u b b b

L u b b b

A

Solution(Method II):

Remark:

1 2 1 2Let , ,..., and , ,..., be two

ordered bases for V

n n

FFE E

E v v v F w w w

S I

: the transition matrix in changing bases from to .

: the matrix representation of the identity operator

with respect and , respectively.

FE

F

E

S E F

I I

E F

1 1 2 2

1

2

1 2

1 2

( ) , 1, 2,...

[ ( )] [ ]

[ ] [ ( )] [ ( )] ... [ ( )]

[ ... ]

j j j j nj j

j

jjj F j F

nj

FE F F n F

n

I v v S w S w S w j n

S

SI v v S

S

I I v I v I v

S S S S

��������������

������������������������������������������

Application I : Computer Graphics and Animation

Fundamental operators: Dilations and Contractions: Reflection about :

e.g., : a reflection about X-axis.

: a reflection about Y-axis.

0110

0110T

)1,1(

cos sin( )

sin cosL x Ax x

)1,1( )0,0(

( )L x cx

1 0

0 1A

1 0

0 1A

axis2

Rotations:

Translations:

Note: Translation is not linear if Homogeneous

Composition of linear mappings is linear!

cos sin( )

sin cosL x Ax x

( )L x x a

11

22

1 1 1 1

2 2 2 2

1

1 0

0 1

0 0 1 1 1

or 0 1 1 1

xx

xx

a x x a

a x x a

A a x Ax a

0a

coordinate

§4.3 Similarity

1 2 1 2

1 1 2 1 1 2

Let { , ,..., } and { , ,..., } be two ordered bases for .

Let { , ,..., } and { , ,..., } be two ordered bases for .n n

m m

E v v v F u u u V

E w w w F z z z W

V WLv

( )L v

Ec1

1Ec

Fc 1

1Fc

[ ]

n

Ev

1[ ( )]

m

EL v

[ ]

n

Fv

1[ ( )]

m

FL v

1[ ]EEA L

1[ ]FFB L

FES

1

1

EFT

coordinate mapping

(transition matrix)

Question:

1

1. How to characterize the relationship between and ?

2. How to choose bases and such that is as simple as

possible like a diagonal matrix ?

A B

E E A

Example:

2 2

1 1

2 1 2

21 2 1 2

Let :

2 ( )

1 1Let [ , ] and [ , ] [ , ] be two ordered bases for R

2 1

1. Find [ ] , [ ]

2. Find the relE F

L R R

x xx L x

x x x

E e e F u u

A L B L

1 2ationship between and in term of [ , ]A B U u u

Solution:

1 1 2

2 1 2

1 1 1 2 1 2

2

21. ( ) 2 1

1 2 0 and ( ) ( )

1 10 ( ) 0 1

1

2 0 1 2 2 ( ) 2 0

1 1 1 2 0

( )

E E E

L e e e

A L L x L x A x Ax

L e e e

L u Au u u u u

L u Au

2 1 2 1 2

1 11 2

1 1 1 11 2 1 2

2 0 1 2 11 1

1 1 1 0 1

2 -1 and

0 1

2 1

0 1

2.

where is

F

u u u u

U Au U Au

B L

B U Au U Au U A u u U AU

U

1 2 1 2

the transition matrix corresponding to the change of basis

from [ , ] to [ , ]F u u E e e

Thm 4.3.1

1 2 1 2

EF

Let , , , and , , , be two

ordered bases for a vector space V and let L be a linear operator

mapping V into itself.

Let S be the transition matrix representing the change from E to

n nE v v v F w w w

1

F.

If and , then F EE FE F

A L B L B S AS

Proof

( ) ( ) ( ) ( )

1

Let

( ) (i)

( ) (ii)

(iii)

( ) ( ) (iv)

( ) ( )

for a

E E

F F

EFE F

EFE F

ii iv i iiiE E EF F FF F E E F

F EE FF F

v V

L v A v

L v B v

v S v

L v S L v

S B v S L v L v A v AS v

B v S AS v

1

ll n

F

F EE F

v

B S AS

A

1

B

1 2

1 2

( )

( )

Since for :

we have ( ) ( ) ( )

E E

FEEF

F F

E

nF E E E

En FE E E

v L v

S S

v L v

I V V

v v

I I w I w I w

w w w S

DEFINITION:

1

Let A and B be n n matrices. B is said to be

similar to A if there exists a nonsingular matrix

such that

S

B S AS

Remark:

1. A is similar to A

2. A is similar to B B is similar to A

3. A is similar to B and B is similar to C

A is similar to C

4. Let and where is a linear operator

A and B are similar

5.

E FA L B L L

11 2

1 2

1 1 2 2

Let ,where , , , , and for some

nonsingular matrix

where , , ,

and 1, 2, ,

nE

nF

j j j nj n

A L E v v v B S AS

S

B L F w w w

w S v S v S v j n

Example1:

3 3

2

2 2

:

2

1, 2 ,4 2 and 1, ,

, , and .FEE F

Let D P P

p a bx cx p b cx

Let E x x F x x

Find A D B D S

Solution:

2

2

2 2

2

2

2

(1) 0 0 1 0 0 0 1 0

( ) 1 1 1 0 0 0 0 2

( ) 2 0 1 2 0 0 0 0

and (1) 0 0 1 0 2 0 (4 2)

(2 ) 2 2 1 0 2 0 (4 2)

(4 2) 8 0 1 4 2

F

D x x

D x x x B D

D x x x x

D x x

D x x x

D x x x

2

1 1

0 (4 2)

0 2 0

0 0 4

0 0 0

1 0 2 1 0 1/ 2

0 2 0 0 1/ 2 0

0 0 4 0 0 1/ 4

F

E EF FF FE E

x

A D

S S A S BS

1 2 3

1 1 1 2 3

2 2 2 1 2 3

3 3 3 1 2 3

let { , , } [ ]

( ) 0 0 0 0

( ) 0 1 0

( ) 4 0 0 4

0 0 0

[ ] 0 1 0

0 0 4

Let

E

F

E e e e A L

L y Ay y y y

L y Ay y y y y

L y Ay y y y y

D L

11 2 3 =[ , , ] E

FS y y y D S AS

3 3

1 2 3

2 2 0

: is defined by ( ) , where 1 1 2

1 1 2

Find the matrix representation [ ] ,

1 2 1

where { , , } 1 , 1 , 1

0 1 1

F

L L x Ax A

D L

F y y y

Example2:

Solution:

Remark:

If the operator can be represented by a diagonal matrix, that

is usually the preferred representation. The prolem of finding a

diagonal representation for a linear operator will be studied in

Chapter 6.