Linear-Time Computation of Similarity Measures for Sequential Data
Chapter 4 Linear Transformations. Outlines Definition and Examples Matrix Representation of linear...
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Transcript of Chapter 4 Linear Transformations. Outlines Definition and Examples Matrix Representation of linear...
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.