Chapter 6
Linear Transformations
6.1 Introduction to Linear Transformations
6.2 The Kernel and Range of a Linear Transformation
6.3 Matrices for Linear Transformations
6.4 Transition Matrices and Similarity
Elementary Linear Algebra
R. Larsen et al. (6 Edition)
6.1 Introduction to Linear Transformations

Function T that maps a vector space V into a vector space W:
T :V  
 W ,
mapping
V , W : vector space
V: the domain of T
W: the codomain of T
Elementary Linear Algebra: Section 6.1, pp.361-362
1/78

Image of v under T:
If v is in V and w is in W such that
T (v)  w
Then w is called the image of v under T .


the range of T:
The set of all images of vectors in V.
the preimage of w:
The set of all v in V such that T(v)=w.
Elementary Linear Algebra: Section 6.1, p.361
2/78

Ex 1: (A function from R2 into R2 )
T :R  R
2
2
v  ( v1 , v 2 )  R
2
T ( v1 , v 2 )  ( v1  v 2 , v1  2 v 2 )
(a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11)
Sol:
( a ) v  (  1, 2 )
 T ( v )  T (  1, 2 )  (  1  2 ,  1  2 ( 2 ))  (  3 , 3 )
( b ) T ( v )  w  (  1, 11 )
T ( v1 , v 2 )  ( v1  v 2 , v1  2 v 2 )  (  1, 11 )
 v1  v 2   1
v1  2 v 2  11
 v1  3 , v 2  4
Thus {(3, 4)} is the preimage of w=(-1, 11).
Elementary Linear Algebra: Section 6.1, p.362
3/78

Linear Transformation (L.T.):
V , W ： vector space
T : V  W ： V to W linear tra
nsformatio
(1) T ( u  v )  T ( u )  T ( v ),
( 2 ) T ( c u )  cT ( u ),
n
u, v  V
c  R
Elementary Linear Algebra: Section 6.1, p.362
4/78

Notes:
(1) A linear transformation is said to be operation preserving.
T (u  v )  T (u )  T ( v )
Addition
in V
Addition
in W
T ( c u )  cT ( u )
Scalar
multiplication
in V
Scalar
multiplication
in W
(2) A linear transformation T : V  V from a vector space into
itself is called a linear operator.
Elementary Linear Algebra: Section 6.1, p.363
5/78

Ex 2: (Verifying a linear transformation T from R2 into R2)
T ( v1 , v 2 )  ( v1  v 2 , v1  2 v 2 )
Pf:
u  ( u1 , u 2 ), v  ( v1 , v 2 ) : vector in R , c : any real number
2
(1)Vector
addition
:
u  v  ( u 1 , u 2 )  ( v1 , v 2 )  ( u 1  v1 , u 2  v 2 )
T ( u  v )  T ( u 1  v1 , u 2  v 2 )
 (( u 1  v1 )  ( u 2  v 2 ), ( u 1  v1 )  2 ( u 2  v 2 ))
 (( u 1  u 2 )  ( v1  v 2 ), ( u 1  2 u 2 )  ( v1  2 v 2 ))
 ( u 1  u 2 , u 1  2 u 2 )  ( v1  v 2 , v1  2 v 2 )
 T (u )  T ( v )
Elementary Linear Algebra: Section 6.1, p.363
6/78
( 2 ) Scalar multiplica tion
c u  c ( u 1 , u 2 )  ( cu 1 , cu 2 )
T ( c u )  T ( cu 1 , cu 2 )  ( cu 1  cu 2 , cu 1  2 cu 2 )
 c (u 1  u 2 , u 1  2 u 2 )
 cT ( u )
Therefore, T is a linear transformation.
Elementary Linear Algebra: Section 6.1, p.363
7/78

Ex 3: (Functions that are not linear transformations)
( a ) f ( x )  sin x
sin( x1  x 2 )  sin( x1 )  sin( x 2 )  f ( x)  sin x is not
sin( 2  3 )  sin( 2 )  sin( 3 )
linear tra nsformatio n
(b ) f ( x )  x
2
2
2
( x1  x 2 )  x1  x 2
2
(1  2 )  1  2
2
2
2
 f ( x)  x is not linear
2
tra nsformatio n
(c ) f ( x )  x  1
f ( x1  x 2 )  x1  x 2  1
f ( x 1 )  f ( x 2 )  ( x 1  1)  ( x 2  1)  x 1  x 2  2
f ( x1  x 2 )  f ( x1 )  f ( x 2 )  f ( x)  x  1 is not
Elementary Linear Algebra: Section 6.1, p.363
linear tra nsformatio n 8/78

Notes: Two uses of the term “linear”.
(1) f ( x )  x  1 is called a linear function because its graph
is a line.
(2) f ( x )  x  1 is not a linear transformation from a vector
space R into R because it preserves neither vector
addition nor scalar multiplication.
Elementary Linear Algebra: Section 6.1, p.364
9/78

Zero transformation:
T :V  W

T ( v )  0,  v  V
Identity transformation:
T :V  V

T (v)  v, v  V
Thm 6.1: (Properties of linear transformations)
T :V  W , u, v  V
(1) T ( 0 )  0
(2) T (  v )   T ( v )
(3) T ( u  v )  T ( u )  T ( v )
(4) If v  c1 v1  c 2 v 2    c n v n
Then T ( v )  T ( c1 v1  c 2 v 2    c n v n )
 c1T ( v1 )  c 2 T ( v 2 )    c n T ( v n )
Elementary Linear Algebra: Section 6.1, p.365
10/78

Ex 4: (Linear transformations and bases)
Let T : R 3  R 3 be a linear transformation such that
T (1, 0 , 0 )  ( 2 ,  1, 4 )
T ( 0 ,1, 0 )  (1,5 ,  2 )
T ( 0 , 0 ,1)  ( 0 ,3,1)
Find T(2, 3, -2).
Sol:
( 2 ,3,  2 )  2 (1, 0 , 0 )  3 ( 0 ,1, 0 )  2 ( 0 , 0 ,1)
T ( 2 ,3 ,  2 )  2 T (1, 0 , 0 )  3T ( 0 ,1, 0 )  2 T ( 0 , 0 ,1)
(T is a L.T.)
 2 ( 2 ,  1, 4 )  3 (1,5 ,  2 )  2 T ( 0 ,3 ,1)
 ( 7 ,7 ,0 )
Elementary Linear Algebra: Section 6.1, p.365
11/78

Ex 5: (A linear transformation defined by a matrix)
The function T : R 2  R 3 is defined
 3
as T ( v )  A v   2

 1
(a) Find T ( v ) , where v  ( 2 ,  1)
2
3
(b) Show that T is a linear tra nsformatio n form R into R
Sol: ( a ) v  ( 2 ,  1)
 3
T (v)  Av   2

 1
0 
 v1 

1  
 v2 
 2
2
R vector R 3 vector
0 
6 
 2   

1
 3


 1
 
 2    0 
 T ( 2 ,  1)  ( 6 ,3 , 0 )
( b ) T ( u  v )  A ( u  v )  A u  A v  T ( u )  T ( v ) (vector addition)
T ( c u )  A ( c u )  c ( A u )  cT ( u )
Elementary Linear Algebra: Section 6.1, p.366
(scalar multiplication)
12/78

Thm 6.2: (The linear transformation given by a matrix)
Let A be an mn matrix. The function T defined by
T (v)  Av
is a linear transformation from Rn into Rm.

Note:
n
R vector
 a11
a
A v   21


 a m 1
a12
a 22



am2

m
R vector
a1 n   v1   a11 v1  a12 v 2    a1 n v n 
a 2 n   v 2   a 21 v1  a 22 v 2    a 2 n v n 
   




  

a mn   v n   a m 1 v1  a m 2 v 2    a mn v n 
T (v)  Av
T :R 
 R
n
m
Elementary Linear Algebra: Section 6.1, p.367
13/78

Ex 7: (Rotation in the plane)
Show that the L.T. T : R 2  R 2 given by the matrix
 cos 
A
 sin 

 sin  
cos  
has the property that it rotates every vector in R2
counterclockwise about the origin through the angle .
Sol:
v  ( x , y )  ( r cos  , r sin  )
(polar coordinates)
r： the length of v
：the angle from the positive
x-axis counterclockwise to
the vector v
Elementary Linear Algebra: Section 6.1, p.368
14/78
 cos 
T (v)  Av 
 sin 

 sin    x   cos 

cos    y   sin 
 sin    r cos  
cos    r sin  
 r cos  cos   r sin  sin  

 r sin  cos   r cos  sin  


 r cos(    ) 

 r sin(    ) 


r：the length of T(v)
 +：the angle from the positive x-axis counterclockwise to
the vector T(v)
Thus, T(v) is the vector that results from rotating the vector v
counterclockwise through the angle .
Elementary Linear Algebra: Section 6.1, p.368
15/78

Ex 8: (A projection in R3)
The linear transformation T : R 3  R 3 is given by
1
A  0

0
0
1
0
0
0

0
is called a projection in R3.
Elementary Linear Algebra: Section 6.1, p.369
16/78

Ex 9: (A linear transformation from Mmn into Mn m )
T ( A)  A
T
(T : M m  n  M n  m )
Show that T is a linear transformation.
Sol:
A, B  M m n
T ( A  B )  ( A  B )  A  B  T ( A)  T ( B )
T ( cA )  ( cA )  cA
T
T
T
T
 cT ( A )
T
Therefore, T is a linear transformation from Mmn into Mn m.
Elementary Linear Algebra: Section 6.1, p.369
17/78
Keywords in Section 6.1:

function: 函數

domain: 論域

codomain: 對應論域

image of v under T: 在T映射下v的像

range of T: T的值域

preimage of w: w的反像

linear transformation: 線性轉換

linear operator: 線性運算子

zero transformation: 零轉換

identity transformation: 相等轉換
18/78
6.2 The Kernel and Range of a Linear Transformation

Kernel of a linear transformation T:
Let T : V  W be a linear transformation
Then the set of all vectors v in V that satisfy T ( v )  0 is
called the kernel of T and is denoted by ker(T).
ker( T )  { v | T ( v )  0 ,  v  V }

Ex 1: (Finding the kernel of a linear transformation)
T ( A)  A
T
(T : M 3 2  M 2  3 )
Sol:
 0

ker( T )   0

 0

0

0 

0  

Elementary Linear Algebra: Section 6.2, p.375
19/78

Ex 2: (The kernel of the zero and identity transformations)
(a) T(v)=0 (the zero transformation T : V  W )
ker( T )  V
(b) T(v)=v (the identity transformation T : V  V )
ker( T )  {0}

Ex 3: (Finding the kernel of a linear transformation)
T ( x , y , z )  ( x , y ,0 )
(T : R  R )
3
3
ker( T )  ?
Sol:
ker( T )  {( 0 , 0 , z ) | z is a real number }
Elementary Linear Algebra: Section 6.2, p.375
20/78

Ex 5: (Finding the kernel of a linear transformation)
 1
T (x)  Ax 
 1

1
2
ker( T )  ?
 x1 
 2  
x2

3  
 x3 
(T : R  R )
3
2
Sol:
ker( T )  {( x1 , x 2 , x 3 ) | T ( x1 , x 2 , x 3 )  ( 0 , 0 ), x  ( x1 , x 2 , x 3 )  R }
3
T ( x1 , x 2 , x 3 )  ( 0 , 0 )
 1
 1

1
2
 x1 
 2    0 
x2 

3     0 
 x3 
Elementary Linear Algebra: Section 6.2, p.376
21/78
1
 1

1
2
2
3
0  G .J .E 1
  

0
0

0
1
1
1
0
0 
 x1   t 
 1 
  x 2     t   t   1
   
 
x
t
 3  
 1 
 ker( T )  {t (1,  1,1) | t is a real number }
 span{( 1,  1,1)}
Elementary Linear Algebra: Section 6.2, p.377
22/78

Thm 6.3: (The kernel is a subspace of V)
The kernel of a linear transformation T : V  W is a
subspace of the domain V.
 T ( 0 )  0 ( Theorem
Pf:
6. 1)
 ker( T ) is a nonempty
Let u and v be vectors
in the kernel of T . then
T (u  v )  T (u )  T ( v )  0  0  0
 u  v  ker( T )
T ( c u )  cT ( u )  c 0  0
 c u  ker( T )
Thus,

subset of V
ker( T ) is a subspace
of V .
Note:
The kernel of T is sometimes called the nullspace of T.
Elementary Linear Algebra: Section 6.2, p.377
23/78

Ex 6: (Finding a basis for the kernel)
Let T : R  R be defined
5
 1

2

A
 1

 0
4
2
0
1
1
3
1
0
2
0
0
0
2
by T ( x )  A x , where x is in R and
5
 1

0

1 

8 
Find a basis for ker(T) as a subspace of R5.
Elementary Linear Algebra: Section 6.2, p.377
24/78
Sol:
A
 1
 2

1

 0
0 
2
1
0
3
1
1
1
0
0
0
2
0
0
2
1
8
0
1
0  G .J .E 0
    
0
0


0 
 0
0
1
2
1
0
0
1
2
0
0
0
0
1
0
4
0
s
0
0

0

0 
t
 x1    2 s  t 
 2
 1 
 x   s  2t 
 1 
 2 
2
  





x  x3 
 s 1 t 0
s
  





 x4    4t 
 0 
 4
 x 5  

 0 
 1 
t
B  (  2 , 1, 1, 0 , 0 ), (1, 2 , 0 ,  4 , 1)  : one basis for the kernel of T
Elementary Linear Algebra: Section 6.2, p.378
25/78

Corollary to Thm 6.3:
Let T : R  R
n
Then the
m
be the L.T given by T ( x )  A x
kernel of T is equal to the solution
space of A x  0
T ( x )  A x ( a linear tr ansformati on T : R  R )
n
 Ker (T )  NS ( A )  x | A x  0 ,  x  R

m
m
 ( subspace
m
of R )
Range of a linear transformation T:
Let T : V  W be a L.T.
Then the
set of all vectors w in W that
are images
of vector
in V is called the range of T and is denoted by range (T )
range (T )  {T ( v ) |  v  V }
Elementary Linear Algebra: Section 6.2, p.378
26/78

Thm 6.4: (The range of T is a subspace of W)
The range of a linear tra nsformatio
n T : V  W is a s u b s p ac e o f W .
Pf:
 T ( 0 )  0 ( Thm.6. 1)
 range (T ) is a nonempty
subset of W
Let T ( u ) and T ( v ) be vector in the range of T
T ( u  v )  T ( u )  T ( v )  range (T )
(u V , v V  u  v V )
T ( c u )  cT ( u )  range (T )
(u V  cu V )
Therefore,
range (T ) is W subspace .
Elementary Linear Algebra: Section 6.2, p.379
27/78

Notes:
T : V  W is a L.T.
(1) Ker (T ) is subspace
of V
( 2 ) range (T ) is subspace

of W
Corollary to Thm 6.4:
Let T : R  R
n
Then the
m
be the L.T. given by T ( x )  A x
range of T is equal to the column
space of A
 range (T )  CS ( A )
Elementary Linear Algebra: Section 6.2, p.379
28/78

Ex 7: (Finding a basis for the range of a linear transformation)
Let T : R  R be defined
5
 1

2

A
 1

 0
4
2
0
1
1
3
1
0
2
0
0
0
2
by T ( x )  A x , where x is R and
5
 1

0

1 

8 
Find a basis for the range of T.
Elementary Linear Algebra: Section 6.2, p.379
29/78
Sol:
2
0
1
1
 2
1
3
1
A
1 0  2 0

0
0
2
 0
c1 c2 c3 c4
 1
1
0  G .J .E 0
    
1
0


8 
 0
c5
w1
0
1
2
1
0
0
0
0
w2
w3
 1
2 
 B
1
4

0
0 
w4 w5
0
0
 w1 , w 2 , w 4  is a basis for CS ( B )
c , c
1
2
, c 4  is a basis for CS ( A )
 (1, 2 ,  1, 0 ), ( 2 , 1, 0 , 0 ), (1, 1, 0 , 2 )  is a basis for the range of T
Elementary Linear Algebra: Section 6.2, pp.379-380
30/78

Rank of a linear transformation T:V→W:
rank (T )  the dimension

Nullity of a linear transformation T:V→W:
nullity (T )  the dimension

of the range of T
of the kernel of T
Note:
Let T : R  R
n
m
be the L.T. given by T ( x )  A x , then
rank (T )  rank ( A )
nullity (T )  nullity ( A )
Elementary Linear Algebra: Section 6.2, p.380
31/78

Thm 6.5: (Sum of rank and nullity)
Let T : V  W be a L.T. form an n - dimensiona
l vector space V
into a vector space W . then
rank (T )  nullity (T )  n
Pf:
dim( range of T )  dim( kernel of T )  dim( domain
of T )
Let T is represente d by an m  n matrix A
Assume
rank ( A )  r
(1) rank (T )  dim( range of T )  dim( column space of A )
 rank ( A )  r
( 2 ) nullity (T )  dim( kernel of T )  dim( solution
space of A )
nr
 rank (T )  nullity (T )  r  ( n  r )  n
Elementary Linear Algebra: Section 6.2, p.380
32/78

Ex 8: (Finding the rank and nullity of a linear transformation)
Find the rank and nullity
1

A 0

 0
0
1
0
of the L.T. T : R  R define by
3
3
 2

1

0 
Sol:
rank (T )  rank ( A )  2
nullity (T )  dim( domain
Elementary Linear Algebra: Section 6.2, p.381
of T )  rank (T )  3  2  1
33/78

Ex 9: (Finding the rank and nullity of a linear transformation)
Let T : R  R be a linear tra nsformatio n.
5
7
( a ) Find the dimension
of the kernel of T if the dimension
of the range is 2
( b ) Find the rank of T if the nullity of T is 4
( c ) Find the rank of T if Ker (T )  { 0}
Sol:
( a ) dim( domain of T )  5
dim( kernel of T )  n  dim( range of T )  5  2  3
( b ) rank (T )  n  nullity (T )  5  4  1
( c ) rank (T )  n  nullity (T )  5  0  5
Elementary Linear Algebra: Section 6.2, p.381
34/78

One-to-one:
A function
T : V  W is called one - to - one if the preimage
every w in the range consists
of
of a single vector.
T is one - to - one iff for all u and v inV, T ( u )  T ( v )
implies
that u  v .
one-to-one
Elementary Linear Algebra: Section 6.2, p.382
not one-to-one
35/78

Onto:
A function
T : V  W is said to be onto if every element
in w has a preimage
in V
(T is onto W when W is equal to the range of T.)
Elementary Linear Algebra: Section 6.2, p.382
36/78

Thm 6.6: (One-to-one linear transformation)
Let T : V  W be a L.T.
Then T is 1 - 1 iff Ker (T )  { 0}
Pf:
Suppose T is 1 - 1
Then T ( v )  0 can have only one solution
:v  0
i.e. Ker (T )  { 0}
Suppose
Ker (T )  { 0} and T ( u )  T ( v )
T (u  v )  T (u )  T ( v )  0
T is a L.T.
 u  v  Ker (T )  u  v  0
 T is 1 - 1
Elementary Linear Algebra: Section 6.2, p.382
37/78

Ex 10: (One-to-one and not one-to-one linear transformation)
( a ) The L.T. T : M m  n  M n  m given by T ( A )  A
T
is one - to - one.
Because its kernel consists of only the m  n zero matrix.
( b ) The zero transform ation T : R  R is not one - to - one.
3
3
3
Because its kernel is all of R .
Elementary Linear Algebra: Section 6.2, p.382
38/78

Thm 6.7: (Onto linear transformation)
Let T : V  W be a L.T., where W is finite dimensiona
l.
Then T is onto iff the rank of T is equal to the dimension

of W .
Thm 6.8: (One-to-one and onto linear transformation)
Let T : V  W be a L.T. with vect or space V and W both of
dimension
n . Then T is one - to - one if and only if it is onto.
Pf:
If T is one - to - one, then Ker (T )  {0} and dim( Ker (T ))  0
dim( range (T ))  n  dim( Ker (T ))  n  dim( W )
Consequent
ly, T is onto.
If T is onto, then dim( range of T )  dim( W )  n
dim( Ker (T ))  n  dim( range of T )  n  n  0
Therefore,
T is one - to - one.
Elementary Linear Algebra: Section 6.2, p.383
39/78

Ex 11:
The L.T. T : R  R
n
of T and determine
1
(a ) A  0

0
1
(c ) A 
0

m
is given by T ( x )  A x , Find the nullity and rank
whether T is one - to - one, onto, or neither.
2
1
0
2
1
0
1

1
1
(b ) A   0

0
0 
 1
1
(d ) A  0

0
Sol:
2
1

0
2
1
0
0
1

0
T:Rn→Rm
dim(domain of T)
rank(T)
nullity(T)
1-1
onto
(a)T:R3→R3
3
3
0
Yes
Yes
(b)T:R2→R3
2
2
0
Yes
No
(c)T:R3→R2
3
2
1
No
Yes
(d)T:R3→R3
3
2
1
No
No
Elementary Linear Algebra: Section 6.2, p.383
40/78

Isomorphism:
A linear tra nsformatio n T : V  W that is one to one and onto
is called an isomorphis m. Moreover, if V and W are vector spaces
such that
there exists an isomorphis m from V to W , then V and W
are said to be isomorphic

to each other.
Thm 6.9: (Isomorphic spaces and dimension)
Two finite-dimensional vector space V and W are isomorphic
if and only if they are of the same dimension.
Pf:
Assume that V is isomorphic
to W , where V has dimension
n.
 There exists a L.T. T : V  W that is one to one and onto.
 T is one - to - one
 dim( Ker (T ))  0
 dim( range of T )  dim( domain of T )  dim( Ker (T ))  n  0  n
Elementary Linear Algebra: Section 6.2, p.384
41/78
 T is onto.
 dim( range of T )  dim( W )  n
Thus dim( V )  dim( W )  n
Assume
that V and W both have dimension
n.
Let v1 , v 2 ,  , v n  be a basis of V, and
let
w1 , w 2 ,  , w n  be a basis of W .
Then an arbitrary vector in V can be represente d as
v  c1 v1  c 2 v 2    c n v n
and you can define a L.T. T : V  W as follows.
T ( v )  c1 w1  c 2 w 2    c n w n
It can be shown that this L.T. is both 1-1 and onto.
Thus V and W are isomorphic.
Elementary Linear Algebra: Section 6.2, p.384
42/78

Ex 12: (Isomorphic vector spaces)
The following vector spaces are isomorphic to each other.
( a ) R  4 - space
4
(b ) M
4 1
 space of all 4  1 matrices
(c ) M
2 2
 space of all 2  2 matrices
( d ) P3 ( x )  space of all polynomial
s of degree 3 or less
( e )V  {( x1 , x 2 , x 3 , x 4 , 0 ), x i is a real number }( subspace
Elementary Linear Algebra: Section 6.2, p.385
5
of R )
43/78
Keywords in Section 6.2:

kernel of a linear transformation T: 線性轉換T的核空間

range of a linear transformation T: 線性轉換T的值域

rank of a linear transformation T: 線性轉換T的秩

nullity of a linear transformation T: 線性轉換T的核次數

one-to-one: 一對一

onto: 映成

isomorphism(one-to-one and onto): 同構

isomorphic space: 同構的空間
44/78
6.3 Matrices for Linear Transformations

Two representations of the linear transformation T:R3→R3 :
(1)T ( x1 , x 2 , x 3 )  ( 2 x1  x 2  x 3 ,  x1  3 x 2  2 x 3 ,3 x 2  4 x 3 )
 2
( 2 )T ( x )  A x    1

 0

1
3
3
 1   x1 
 2  x2 
 
4   x3 
Three reasons for matrix representation of a linear transformation:

It is simpler to write.

It is simpler to read.

It is more easily adapted for computer use.
Elementary Linear Algebra: Section 6.3, p.387
45/78

Thm 6.10: (Standard matrix for a linear transformation)
Let T : R  R
n
m
be a linear trt ansformati
on such that
 a 11 
 a12 
 a1 n 
a 
a 
a 
21
22
2n
T ( e1 )  
 , T (e2 )  
 ,  , T (en )  
,









 a m 1 
 a m 2 
 a mn 
Then the
m  n matrix who
A  T ( e1 )
is such that
T (e2 )

se n columns
 a11
a
T ( e n )    21


 a m 1
T ( v )  A v for every
A is called the standard
Elementary Linear Algebra: Section 6.3, p.388
correspond
a12
a 22




am2

to T ( e i )
a1 n 
a2n 



a mn 
n
v in R .
matrix for T .
46/78
Pf:
 v1 
v 
v   2   v1 e1  v 2 e 2    v n e n

 
 v n 
T is a L.T.  T ( v )  T ( v 1 e1  v 2 e 2    v n e n )
 T ( v 1 e1 )  T ( v 2 e 2 )    T ( v n e n )
 v 1T ( e1 )  v 2 T ( e 2 )    v n T ( e n )
 a11
a
A v   21


 a m 1
a12
a 22




am2

Elementary Linear Algebra: Section 6.3, p.388
a 1 n   v1   a11 v1  a12 v 2    a1 n v n 
a 2 n   v 2   a 21 v1  a 22 v 2    a 2 n v n 
   




  

a mn   v n   a m 1 v1  a m 2 v 2    a mn v n 
47/78
 a 11 
 a 12 
 a1 n 
a 
a 
a 
21
22
2n
 v1 
  v2 
    vn 










 a m 1 
 a m 2 
 a mn 
 v1T ( e1 )  v 2 T ( e 2 )    v n T ( e n )
Therefore,
T ( v )  A v for each v in R
Elementary Linear Algebra: Section 6.3, p.389
n
48/78

Ex 1: (Finding the standard matrix of a linear transformation)
Find the standard
matrix for the L.T. T : R  R define by
3
2
T ( x, y, z )  ( x  2 y, 2 x  y )
Sol:
Vector Notation
T ( e1 )  T (1, 0 , 0 )  (1, 2 )
T ( e 2 )  T ( 0 , 1, 0 )  (  2 , 1)
T ( e 3 )  T ( 0 , 0 , 1)  ( 0 , 0 )
Elementary Linear Algebra: Section 6.3, p.389
Matrix Notation
1 
1 


T ( e1 )  T ( 0 ) 
2
 
 
0
 
0 
 2
T (e2 )  T ( 1  ) 
 1 
 


0 
0 
0 


T ( e3 )  T ( 0 ) 
0 
 
 
1
 
49/78
A  T ( e1 )


1
2

2
1
T (e2 )
T ( e3 ) 
0
0 
Check:
x
1


A y 
   2
z
2
1
x
0    x  2 y 
y 

0     2 x  y 
z
i.e. T ( x , y , z )  ( x  2 y , 2 x  y )

Note:
A
1
2

2
1
0   1x  2 y  0 z
0   2 x  1 y  0 z
Elementary Linear Algebra: Section 6.3, p.389
50/78

Ex 2: (Finding the standard matrix of a linear transformation)
The linear tra nsformatio
n T : R  R is given by projecting
2
2
2
each point in R onto the x - axis. Find the standard
matrix for T .
Sol:
T ( x, y )  ( x, 0)
A  T ( e1 )

T ( e 2 )   T (1, 0 )
T ( 0 , 1)  
1
0

0
0 
Notes:
(1) The standard matrix for the zero transformation from Rn into Rm
is the mn zero matrix.
(2) The standard matrix for the zero transformation from Rn into Rn
is the nn identity matrix In
Elementary Linear Algebra: Section 6.3, p.390
51/78

Composition of T1:Rn→Rm with T2:Rm→Rp :
T ( v )  T 2 (T1 ( v )),
T  T 2  T1 , domain

vR
n
of T  domain
of T1
Thm 6.11: (Composition of linear transformations)
Let T1 : R  R
n
with standard
(1) The compositio
( 2 ) The standard
m
and T 2 : R
matrices
m
 R
matrix
be L.T.
A1 and A2 , then
n T : R  R , defined
n
p
p
by T ( v )  T 2 (T1 ( v )), is a L.T.
A for T is given by the matrix
Elementary Linear Algebra: Section 6.3, p.391
product
A  A 2 A1
52/78
Pf:
(1) ( T is a L.T.)
n
Let u and v be vectors in R and let c be any scalar the n
T ( u  v )  T 2 (T1 ( u  v ))  T 2 (T1 ( u )  T1 ( v ))
 T 2 (T1 ( u ))  T 2 (T1 ( v ))  T ( u )  T ( v )
T ( c v )  T 2 (T1 ( c v ))  T 2 ( cT 1 ( v ))  cT 2 (T1 ( v ))  cT ( v )
( 2 )( A 2 A1 is the standard
matrix
for T )
T ( v )  T 2 (T1 ( v ))  T 2 ( A1 v )  A 2 A1 v  ( A 2 A1 ) v

Note:
T1  T 2  T 2  T1
Elementary Linear Algebra: Section 6.3, p.391
53/78

Ex 3: (The standard matrix of a composition)
3
3
Let T1 and T 2 be L.T. from R into R s.t.
T1 ( x , y , z )  ( 2 x  y , 0 , x  z )
T2 ( x , y , z )  ( x  y , z , y )
Find the standard
Sol:
matrices
for the compositio
ns
T  T 2  T1 and T '  T1  T 2 ,
2

A1  0

 1
1
1

A2  0

 0
1
0
0
0
1
0

0 ( standard

1 
0

1 ( standard

0 
Elementary Linear Algebra: Section 6.3, p.392
matrix for T1 )
matrix for T 2 )
54/78
The standard
matrix
1
A  A 2 A1   0

0
The standard
matrix
2

A '  A1 A 2  0

 1
for T  T 2  T1
1
0
1
0 2
1  0

0  1
1
0
0
0 2
0   1
 
1  0
1
0 2
 
1  0
 
0   1
2
0
0
0
1

0
for T '  T1  T 2
1
0
0
Elementary Linear Algebra: Section 6.3, p.392
0  1

0 0

1   0
1
0
1
0
0
1

0

0 
55/78

Inverse linear transformation:
If T1 : R  R and T 2 : R  R are L.T. s.t. for every v in R
n
n
n
n
T 2 (T1 ( v ))  v and
Then T 2 is called the inverse

n
T1 (T 2 ( v ))  v
of T1 and T1 is said to be invertible
Note:
If the transformation T is invertible, then the inverse is
unique and denoted by T–1 .
Elementary Linear Algebra: Section 6.3, p.392
56/78

Thm 6.12: (Existence of an inverse transformation)
Let T : R  R be a L.T. with standard
n
Then the
n
following
condition
matrix
are equivalent
A,
.
(1) T is invertible.
(2) T is an isomorphism.
(3) A is invertible.

Note:
If T is invertible with standard matrix A, then the standard
matrix for T–1 is A–1 .
Elementary Linear Algebra: Section 6.3, p.393
57/78

Ex 4: (Finding the inverse of a linear transformation)
The L.T. T ： R  R is defined
3
3
by
T ( x1 , x 2 , x 3 )  ( 2 x1  3 x 2  x 3 , 3 x1  3 x 2  x 3 , 2 x1  4 x 2  x 3 )
Show that T is invertible, and find its inverse.
Sol:
The standard
2

A 3

 2
A
3
3
4
matrix for T
 2 x1  3 x2  x3
1

1

1
2
I3   3

2
 3 x1  3 x2  x3
 2 x1  4 x2  x3
3
1
1
0
3
1
0
1
4
1
0
0
Elementary Linear Algebra: Section 6.3, p.393
0
0

1
58/78
1
G .J .E
    0

0
Therefore
A
1
 1
  1

 6
0
0
1
1
1
0
1
0
0
1
6
2
T is invertible
1
0
2
1

and the standard
A
1

matrix for T
1
is A
1
0 
1 

 3
 1
1
1
T ( v )  A v   1

 6
In other word s,
T
0 
1  I

 3
1
0
2
0   x1  
 x1  x 2


1   x2   
 x1  x 3
  

 3   x 3   6 x1  2 x 2  3 x 3 
( x 1 , x 2 , x 3 )  (  x 1  x 2 ,  x1  x 3 , 6 x 1  2 x 2  3 x 3 )
Elementary Linear Algebra: Section 6.3, p.394
59/78

the matrix of T relative to the bases B and B':
T :V  W
B  {v1 , v 2 ,  , v n }
( a L.T. )
(a basis for V )
B '  { w 1 , w 2 ,  , w m } (a basis for W )
Thus, the matrix of T relative to the bases B and B' is
A  T ( v1 ) B ' , T ( v 2 ) B ' ,  , T ( v n ) B '   M m  n
Elementary Linear Algebra: Section 6.3, p.394
60/78

Transformation matrix for nonstandard bases:
Let V and W be finite - dimensiona
l vector spaces with basis B and B ' ,
respective ly, where B  { v 1 , v 2 ,  , v n }
If T : V  W is a L.T. s.t.
T ( v1 ) B '
then the
 a 11 
a 
  21  ,



 a m 1 
T ( v 2 ) B '
m  n matrix who
 a 12 
a 
  22  ,  ,



 a m 2 
se n columns
Elementary Linear Algebra: Section 6.3, p.395
T ( v n ) B '
correspond
 a1 n 
a 
  2n 



 a mn 
to T ( v i ) B '
61/78
A  T ( e1 )
is such that
T (e2 )

T ( v ) B ' 
Elementary Linear Algebra: Section 6.3, p.395
 a11
a
T ( e n )    21


 a m 1
a12
a 22




am2

a1 n 
a2n 



a mn 
A[ v ] B for every v in V .
62/78

Ex 5: (Finding a matrix relative to nonstandard bases)
Let T ： R  R be a L.T. defined
2
2
by
T ( x1 , x 2 )  ( x1  x 2 , 2 x1  x 2 )
Find the matrix of T relative
to the basis
B  {( 1, 2 ), (  1, 1)} and B '  {( 1, 0 ), ( 0 , 1)}
Sol:
T (1, 2 )  ( 3 , 0 )  3 (1, 0 )  0 ( 0 , 1)
T (  1, 1)  ( 0 ,  3 )  0 (1, 0 )  3 ( 0 , 1)
T (1, 2 ) B '  
3
,

0 
the matrix
T (  1, 1) B '  
0 

  3
for T relative
A  T (1, 2 ) B '
Elementary Linear Algebra: Section 6.3, p.395
to B and B '
T (1, 2 ) B '   
3
0
0 
 3 
63/78

Ex 6:
For the L.T. T ： R  R given in Example
2
2
5, use the matrix
A
to find T ( v ), where v  ( 2 , 1)
Sol:
v  ( 2 , 1)  1(1, 2 )  1(  1, 1)
  v B 
B  {(1, 2), (1, 1)}
1 
  1
 
 T ( v ) B '  A  v B
3

0

0   1  3




 3    1  3 
 T ( v )  3 (1, 0 )  3 ( 0 , 1)  ( 3 , 3 )

B'  {(1, 0), (0, 1)}
Check:
T ( 2 , 1)  ( 2  1, 2(2)  1)  ( 3 , 3 )
Elementary Linear Algebra: Section 6.3, p.395
64/78

Notes:
(1)In the
special case where V  W and B  B ' ,
the matrix
A is alled the matrix
( 2 )T : V  V : the identity t
of T relative
ransformat
to the basis B
ion
B  { v1 , v 2 ,  , v n } : a basis for V
 the matrix of T relative
to the basis B
1

0

A  T ( v1 ) B , T ( v 2 ) B ,  , T ( v n ) B  


0
Elementary Linear Algebra, Section 6.3, p.396
0

1



0

0

0
 I
n


1
65/78
Keywords in Section 6.3:

standard matrix for T: T 的標準矩陣

composition of linear transformations: 線性轉換的合成

inverse linear transformation: 反線性轉換


matrix of T relative to the bases B and B' : T對應於基底B到
B'的矩陣
matrix of T relative to the basis B: T對應於基底B的矩陣
66/78
6.4 Transition Matrices and Similarity
T :V  V
( a L.T. )
B  { v1 , v 2 ,  , v n }
( a basis of V )
B '  { w1 , w 2 ,  , w n } (a basis of V )
A  T ( v1 ) B , T ( v 2 ) B ,  , T ( v n ) B 
( matrix of T relative
to B )
A '  T ( w1 ) B ' , T ( w 2 ) B ' ,  , T ( w n ) B ' 
(matrix
to B ' )
P  w1 B , w 2 B ,  , w n B 
P
1
 v1 B ' , v 2 B ' ,  , v n B ' 
 v B  P v B ' , v B '  P
1
of T relative
( transitio
n matrix from B ' to B )
( transitio
n matrix from B to B ' )
v B
T ( v ) B  A v B
T ( v ) B '  A ' v B '
Elementary Linear Algebra: Section 6.4, p.399
67/78

Two ways to get from v B ' to T ( v)B ':
indirect
(1)( direct )
A ' [ v ] B '  [T ( v )] B '
( 2 ) (indirect)
1
P AP [ v ] B '  [T ( v )] B '
 A'  P
1
direct
AP
Elementary Linear Algebra: Section 6.4, pp.399-400
68/78

Ex 1: (Finding a matrix for a linear transformation)
Find the matrix
A' for T ： R  R
2
2
T ( x1 , x 2 )  ( 2 x1  2 x 2 ,  x1  3 x 2 )
reletive
Sol:
to the basis B '  {( 1, 0 ), (1, 1)}
( I) A '  T (1, 0 ) B '
T (1, 1) B ' 
T (1, 0 )  ( 2 ,  1)  3 (1, 0 )  1(1, 1) 
T (1, 0 ) B '  
T (1, 1)  ( 0 , 2 )   2 (1, 0 )  2 (1, 1) 
T (1, 1) B '  
 A '  T (1, 0 ) B '
Elementary Linear Algebra: Section 6.4, p.400
 3
T (1, 1) B '   
 1
3 

  1
 2

 2 
 2
2 
69/78
( II) standard
matrix for T ( matrix of T relative
A  T (1, 0 )
transition
T ( 0 , 1)  
matrix
to B  {( 1, 0 ), ( 0 , 1)})
 2
3 
 2
 1

from B ' to B
 1 1
P  (1, 0 ) B (1, 1) B   

0
1


transition matrix from B to B '
P
1
1
 
0
matrix
A'  P
 1

1 
of T relative
1
1
AP  
0
B'
 1  2

1   1
Elementary Linear Algebra: Section 6.4, p.400
 2  1

3  0
1  3
 
1   1
 2

2 
70/78

Ex 2: (Finding a matrix for a linear transformation)
Let B  {(  3 , 2 ), ( 4 ,  2 )} and B '  {(  1, 2 ), ( 2 ,  2 )} be basis for R ,
2
 2
and let A  
 3
7
2
2
be
the
matrix
for
T
:
R

R
relative

7
Find the matrix of T relative
Sol:
transition
transition
to B '.
matrix from B ' to B : P  (  1, 2 ) B
matrix from B to B ' : P
1
 (  3, 2 ) B '
matrix of T relative to B ' :
A'  P
1
1
AP  
 2
Elementary Linear Algebra: Section 6.4, p.401
to B .
2  2
3    3
7  3
7   2
3
( 2 ,  2 ) B   
2
 2

 1
1
( 4 ,  2 ) B '   
 2
 2  2
 

 1   1
2

3
1
3 
71/78

Ex 3: (Finding a matrix for a linear transformation)
For the linear tra nsformatio
n T : R  R given in Ex.2, find
T ( v ) B and T ( v ) B ' , for the
vector v whose coordinate
2
 v B '
Sol:
 v B
 P  v B '
T ( v ) B
3

2

 A  v B
T ( v ) B ' 
P
1
2
 v B、
matrix is
  3
 


1


 2    3   7 




 1    1    5 
 2

 3

T ( v ) B
or T ( v ) B '  A '  v B '
7    7    21 




7    5    14 
1

 2

 2

 1
Elementary Linear Algebra: Section 6.4, p.401
2    21    7 




3    14   0 
1    3   7 



3   1  0 
72/78


Similar matrix:
For square matrices A and A‘ of order n, A‘ is said to be
similar to A if there exist an invertible matrix P s.t. A '  P  1 AP
Thm 6.13: (Properties of similar matrices)
Let A, B, and C be square matrices of order n.
Then the following properties are true.
(1) A is similar to A.
(2) If A is similar to B, then B is similar to A.
(3) If A is similar to B and B is similar to C, then A is similar to C.
Pf:
(1) A  I n AI n
1
( 2 ) A  P BP
PAP
1
 B

PAP

Q
Elementary Linear Algebra: Section 6.4, p.402
1
1
1
 P ( P BP ) P
1
1
AQ  B ( Q  P )
73/78

Ex 4: (Similar matrices)
 2
(a ) A  
 1
because
 2
(b ) A  
 3
because
 2
 and A ' 
3 
 3

 1
 2
 are similar
2 
1
A '  P AP , where P  
0
1
7
 and A ' 
7
 2

 1
1
 are similar
3
3
A '  P AP , where P  
2
1
Elementary Linear Algebra: Section 6.4, p.403
1

1
 2

 1
74/78

Ex 5: (A comparison of two matrices for a linear transformation)
Suppose
1

A 3

 0
to the standard
3
1
0
0 

3
3
0 is the matrix for T : R  R relative

 2 
basis. Find the matrix for T relative
to the basis
B '  {( 1, 1, 0 ), (1,  1, 0 ), ( 0 , 0 , 1)}
Sol:
The transitio n matrix from B' to the standard matrix
P  (1, 1, 0 ) B
 P
1

 

0
1
2
1
2
(1,  1, 0 ) B
1
2

0
1
2
0
0

1
Elementary Linear Algebra: Section 6.4, p.403
1
 ( 0 , 0 , 1)  B    1

 0
1
1
0
0
0

1 
75/78
matrix of T relative
 12

1
A '  P AP  12

 0
4

 0

 0
0
2
0
to B ' :
1
2

1
2
0
0  1

0 3

1   0
3
1
0
0  1

0
1

 2   0
1
1
0
0

0

1 
0 

0

 2 
Elementary Linear Algebra: Section 6.4, p.403
76/78

Notes: Computational advantages of diagonal matrices:
(1) D
k
(2) D
(3) D
 d 1k
 0
 
 
 0
T
1
0
k

d2


0


0 
0 

 
k
d n 
d1
0
D 


 0
0
d2




0

0 
0 



d n 
D
 d11
0
 

0

0

1
d2


0


Elementary Linear Algebra: Section 6.4, p.404
0
0
, d i  0

1 
dn 
77/78
Keywords in Section 6.4:

matrix of T relative to B: T 相對於B的矩陣

matrix of T relative to B' : T 相對於B'的矩陣

transition matrix from B' to B : 從B'到B的轉移矩陣

transition matrix from B to B' : 從B到B'的轉移矩陣

similar matrix: 相似矩陣
78/78