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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.