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.