### Linear Transformations

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 mapping
W ,
V ,W : vectorspace
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 2  R 2 v  (v1 , v2 )  R 2
T (v1 , v2 )  (v1  v2 , v1  2v2 )
(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 , v2 )  (v1  v2 , v1  2v2 )  (1, 11)
 v1  v2  1
v1  2v2  11
 v1  3, v2  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： vect orspace
T : V  W： V toW linear transformation
(1) T (u  v)  T (u)  T ( v), u, v V
(2) T (cu)  cT (u),
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)
in V
in W
T (cu)  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 , v2 )  (v1  v2 , v1  2v2 )
Pf:
u  (u1, u2 ), v  (v1, v2 ) : vectorin R2 , c : any real number
u  v  (u1 , u2 )  (v1 , v2 )  (u1  v1 , u2  v2 )
T (u  v )  T (u1  v1 , u 2  v2 )
 ((u1  v1 )  (u2  v2 ), (u1  v1 )  2(u2  v2 ))
 ((u1  u2 )  (v1  v2 ), (u1  2u2 )  (v1  2v2 ))
 (u1  u2 , u1  2u2 )  (v1  v2 , v1  2v2 )
 T (u)  T ( v )
Elementary Linear Algebra: Section 6.1, p.363
6/78
(2) Scalar multiplication
cu  c(u1 , u2 )  (cu1 , cu 2 )
T (cu)  T (cu1 , cu 2 )  (cu1  cu 2 , cu1  2cu 2 )
 c(u1  u 2 , u1  2u 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  x2 )  sin(x1 )  sin(x2 )  f ( x)  sin x is not
sin(2  3 )  sin(2 )  sin(3 )
(b) f ( x)  x 2
( x1  x2 )2  x12  x22
(1  2)  1  2
2
2
2
linear tra nsformatio n
 f ( x)  x 2 is not linear
tra nsformatio n
(c) f ( x)  x  1
f ( x1  x2 )  x1  x2  1
f ( x1 )  f ( x2 )  ( x1  1)  ( x2  1)  x1  x2  2
f ( x1  x2 )  f ( x1 )  f ( x2 )  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
Elementary Linear Algebra: Section 6.1, p.364
9/78

Zero transformation:
T :V  W

Identity transformation:
T :V  V

T ( v)  0, 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  c1v1  c2v2    cn vn
T henT ( v)  T (c1v1  c2v2    cn vn )
 c1T (v1 )  c2T (v2 )    cnT (vn )
Elementary Linear Algebra: Section 6.1, p.365
10/78

Ex 4: (Linear transformations and bases)
Let T : R3  R3 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)  2T (1,0,0)  3T (0,1,0)  2T (0,0,1)
 2(2,1,4)  3(1,5,2)  2T (0,3,1)
 (7,7,0)
Elementary Linear Algebra: Section 6.1, p.365
(T is a L.T.)
11/78

Ex 5: (A linear transformation defined by a matrix)
0
3
 v1 
2
3


1  
The function T : R  R is defined as T ( v)  Av  2

 v 2 
  1  2
(a) Find T ( v) , where v  (2,1)
(b) Show that T is a linear transformation form R 2 into R3
Sol: (a) v  (2,1)
R 2 vector R 3 vector
0
3
6
2  


T ( v)  Av  2
1   3

  1  

1

2


0
T (2,1)  (6,3,0)
(b) T (u  v)  A(u  v)  Au  Av  T (u)  T ( v)
T (cu)  A(cu)  c( Au)  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:
R n vector
 a11
 a21
Av  
 
am1
R m vector
a12  a1n   v1   a11v1  a12 v2    a1n vn 
a22  a2 n  v2   a21v1  a22 v2    a2 n vn 
   


    


am 2  amn  vn  am1v1  am 2 v2    amn vn 
T ( v)  Av
T : Rn 
 R 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  sin    x  cos
T ( v)  Av  




sin

cos

y

    sin 
r cos cos  r sin  sin  

r sin  cos  r cos sin  
r cos(   )

 r sin(   ) 
 sin   r cos 
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 0 0
A  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)  AT
(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)T  AT  BT  T ( A)  T ( B)
T (cA)  (cA)T  cAT  cT ( A)
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)  AT (T : M32  M 23 )
Sol:
 0 0  



ker(T )  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 : R3  R3 )
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)
 x1 
 1  1  2  
T (x)  Ax  
x2

3  
 1 2
 x3 
ker(T )  ?
(T : R 3  R 2 )
Sol:
ker(T )  {( x1, x2 , x3 ) | T ( x1, x2 , x3 )  (0,0), x  ( x1, x2 , x3 )  R3}
T ( x1 , x2 , x3 )  (0,0)
 x1 
 1  1  2    0 
x2   
 1 2

3    0 
 x3 
Elementary Linear Algebra: Section 6.2, p.376
21/78
 1  1  2 0 G. J .E 1 0  1 0

 1 2

3 0
0 1 1 0
 x1   t   1 
  x2    t   t  1
     
 x3   t   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 (T heorem6.1)
 ker(T ) is a nonemptysubset of V
Pf:
Let u and v be vectors in the kernel of T . then
T (u  v)  T (u)  T ( v)  0  0  0
T (cu)  cT (u)  c0  0
Thus, ker(T ) is a subspace of V .

 u  v  ker(T )
 cu  ker(T )
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 5  R 4 be defined by T (x)  Ax, where x is in R 5 and
1
2
A
 1

0
1  1
1 3 1 0 
0 2 0 1 

0 0 2 8
2
0
Find a basis for ker(T) as a subspace of R5.
Elementary Linear Algebra: Section 6.2, p.377
Sol:
A
1
2

 1
 0
0 
2 0
1 3
0 2
0 0
1 1
1 0
0 1
2 8
0
1
0  G . J . E 0
  
0
0
0
0
0 2
1 1
0 0
0 0
s
0 1
0 2
1 4
0 0
t
0
0

0
0
 x1   2s  t 
  2  1 
 x2   s  2t 
1 2
 
x   x3    s   s  1   t  0 
 x4    4t 
 0    4
 x5   t 
 0   1 
B  (2, 1, 1, 0, 0), (1, 2, 0,  4, 1): one basis for thekernelof T
Elementary Linear Algebra: Section 6.2, p.378
25/78

Corollary to Thm 6.3:
Let T : R n  R m be theL.T given by T (x)  Ax
T hen thekernelof T is equal to thesolutionspace of Ax  0
T (x)  Ax (a linear tr ansformati on T : R n  R m )
 Ker (T )  NS ( A)  x | Ax  0, x  R m  (subspace of R m )

Range of a linear transformation T:
Let T : V  W be a L.T .
T hen theset of all vectorsw in W thatare images of vector
in V is called therange of T and is denotedby 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)
T herange of a linear transformation T : V  W is a subspace of W .
Pf:
T (0)  0 (T hm.6.1)
 range(T ) is a nonemptysubset of W
Let T (u) andT ( v) be vectorin therange of T
T (u  v)  T (u)  T ( v)  range(T ) (u V , v V  u  v V )
T (cu)  cT (u)  range(T )
(u V  cu V )
T herefore,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 n  R m be theL.T .given by T (x)  Ax
T hen therange of T is equal to thecolumnspace 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 5  R 4 be defined by T (x)  Ax, where x is R 5 and
1
2
A
 1

0
1  1
1 3 1 0 
0 2 0 1 

0 0 2 8
2
0
Find a basis for the range of T.
Elementary Linear Algebra: Section 6.2, p.379
29/78
Sol:
1
2
A
 1
 0
2 0
1 3
0 2
0 0
1  1
1
1 0  G . J . E 0
 
0 1
0
2 8 
0
c1 c2 c3 c4 c5
0 2
1 1
0 0
0 0
0  1
0 2
B
1 4
0 0 
w1 w2 w3 w4 w5
 w1 , w2 , w4 is a basis for CS ( B)
c , c , c is a basis for CS ( A)
1
2
4
 (1, 2, 1, 0), (2, 1, 0, 0), (1, 1, 0, 2)is a basis for therangeof T
Elementary Linear Algebra: Section 6.2, pp.379-380
30/78

Rank of a linear transformation T:V→W:
rank(T )  thedimensionof therange of T

Nullity of a linear transformation T:V→W:
nullity(T )  thedimensionof thekernelof T

Note:
Let T : R n  R m be theL.T .given by T (x)  Ax, 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 .forman n - dimensional vectorspaceV
intoa vectorspaceW . then
rank(T )  nullity(T )  n
dim(range of T )  dim(kernelof T )  dim(domain of T )
Pf:

Let T is represented 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(kernelof T )  dim(solutionspace 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 t herank and nullit yof t heL.T .T : R 3  R 3 define by
1 0  2
A  0 1 1 
0 0 0 
Sol:
rank(T )  rank( A)  2
nullity(T )  dim(domain of T )  rank(T )  3  2  1
Elementary Linear Algebra: Section 6.2, p.381
33/78

Ex 9: (Finding the rank and nullity of a linear transformation)
Let T : R 5  R 7 be a linear tra nsformatio n.
(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 functionT : V  W is called one- to - oneif thepreimageof
every win therange consistsof a single vector.
T is one- to - oneiff for all u and v inV, T (u)  T ( v)
implies thatu  v.
one-to-one
Elementary Linear Algebra: Section 6.2, p.382
not one-to-one
35/78

Onto:
A functionT : V  W is said to be ontoif everyelement
in w has a preimagein 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 .
T henT is 1 - 1 iff Ker (T )  {0}
Pf:
SupposeT is 1 -1
T henT (v)  0 can haveonlyonesolution: v  0
i.e. Ker(T )  {0}
Suppose Ker(T )  {0} andT (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)  AT
is one - to - one.
Because its kernel consists of only the m n zero matrix.
(b) The zero transformation T : R 3  R 3 is not one - to - one.
Because its kernel is all of R3 .
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 n  R m is given by T (x)  Ax, Find the nullity and rank
of T and determine whether T is one - to - one, onto, or neither.
1 2 0
(a) A  0 1 1


0 0 1
1 2 0 
(c ) A  
0 1  1
Sol:
1 2 
(b) A  0 1 


0 0 
1 2
( d ) A  0 1

0 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 transformation 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
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 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 , v2 , , vn  be a basis of V, and
let w1 , w2 , , wn  be a basis of W .
Then an arbitrary vector in V can be represente d as
v  c1v1  c2 v2    cn vn
and you can define a L.T. T : V  W as follows.
T ( v )  c1w1  c2 w2    cn wn
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) R4  4 - space
(b)M 41  space of all 4 1 matrices
(c)M 22  space of all 2  2 matrices
(d ) P3 ( x)  spaceof all polynomial
s of degree 3 or less
(e)V  {( x1, x2 , x3 , x4 , 0), xi is a real number}(subspace of R5 )
Elementary Linear Algebra: Section 6.2, p.385
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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
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6.3 Matrices for Linear Transformations

Two representations of the linear transformation T:R3→R3 :
(1)T ( x1, x2 , x3 )  (2x1  x2  x3 , x1  3x2  2x3 ,3x2  4x3 )
 2 1  1  x1 
(2)T (x)  Ax   1 3  2  x2 

 
0
3
4

  x3 

Three reasons for matrix representation of a linear transformation:

It is simpler to write.


It is more easily adapted for computer use.
Elementary Linear Algebra: Section 6.3, p.387
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
Thm 6.10: (Standard matrix for a linear transformation)
Let T : R n  R m be a linear trtansformation such that
 a11 
 a12 
 a1n 
 a21 
 a22 
 a2 n 
T (e1 )   , T (e2 )   ,  , T (en )   ,
  
  
  
am1 
am 2 
amn 
Then them n matrix whose n columnscorrespondtoT (ei )
 a11 a12  a1n 
 a21 a22  a2 n 
A  T (e1 ) T (e2 )  T (en )  







am1 am 2  amn 
is such thatT ( v)  Av for every v in R n .
A is called thestandardmatrixfor T .
Elementary Linear Algebra: Section 6.3, p.388
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Pf:
 v1 
 v2 
v     v1e1  v2 e2    vn en

vn 
T is a L.T . T (v )  T (v1e1  v2 e2    vn en )
 T (v1e1 )  T (v2 e2 )    T (vn en )
 v1T (e1 )  v2T (e2 )    vnT (en )
 a11
 a21
Av  
 
am1
a12
a22

am 2
 a1n   v1   a11v1  a12 v2    a1n vn 
 a2 n  v2   a21v1  a22 v2    a2 n vn 
   

     


 amn  vn  am1v1  am 2 v2    amn vn 
Elementary Linear Algebra: Section 6.3, p.388
47/78
 a11 
 a12 
 a1n 
 a21 
 a22 
 a2 n 
 v1    v2      vn  
  
  
  
am1 
am 2 
amn 
 v1T (e1 )  v2T (e2 )    vnT (en )
T herefore,T ( v)  Av for each v in Rn
Elementary Linear Algebra: Section 6.3, p.389
48/78

Ex 1: (Finding the standard matrix of a linear transformation)
Find thestandardmatrixfor theL.T .T : R3  R2 define by
T ( x, y, z)  ( x  2 y, 2 x  y)
Sol:
Vector Notation
T (e1 )  T (1, 0, 0)  (1, 2)
T (e2 )  T (0, 1, 0)  (2, 1)
T (e3 )  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 ) T (e2 ) T (e3 )
1  2 0

2 1 0

Check:
 x
 x
1  2 0     x  2 y 


A y 
y 

  2 1 0   2 x  y 
z
z
i.e.T ( x, y, z )  ( x  2 y,2 x  y)

Note:
1  2 0  1x  2 y  0 z
A
2 1 0  2 x  1y  0 z
Elementary Linear Algebra: Section 6.3, p.389
50/78

Ex 2: (Finding the standard matrix of a linear transformation)
T helinear transformation T : R 2  R 2 is given by projecting
each pointin R 2 ontothe x - axis. Find thestandardmatrixfor T .
Sol:
T ( x, y)  ( x, 0)
1 0
A  T (e1 ) T (e2 )  T (1, 0) T (0, 1)  
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)  T2 (T1 ( v)), v  Rn
T  T2  T1 , domainof T  domainof T1

Thm 6.11: (Composition of linear transformations)
Let T1 : R n  R m and T2 : R m  R p be L.T .
with standardmatricesA1 and A2 , then
(1)Thecomposition T : Rn  R p , definedby T ( v)  T2 (T1 ( v)), is a L.T.
(2) T hestandardmatrixA forT is given by thematrixproduct A  A2 A1
Elementary Linear Algebra: Section 6.3, p.391
52/78
Pf:
(1)( T is a L.T .)
Let u and v be vectorsin R n and let c be any scalar then
T (u  v)  T2 (T1 (u  v))  T2 (T1 (u)  T1 ( v))
 T2 (T1 (u))  T2 (T1 ( v))  T (u)  T ( v)
T (cv)  T2 (T1 (cv))  T2 (cT1 ( v))  cT2 (T1 ( v))  cT ( v)
(2)( A2 A1 is thestandardmatrixfor T )
T ( v)  T2 (T1 ( v))  T2 ( A1v)  A2 A1v  ( A2 A1 ) v

Note:
T1  T2  T2  T1
Elementary Linear Algebra: Section 6.3, p.391
53/78

Ex 3: (The standard matrix of a composition)
Let T1 andT2 be L.T.fromR3 into R3 s.t.
Sol:
T1 ( x, y, z)  (2 x  y, 0, x  z)
T2 ( x, y, z)  ( x  y, z, y)
Find thestandardmatricesfor thecompositions
T  T2  T1 and T '  T1  T2 ,
2
A1  0
1
1
A2  0
0
1 0
0 0 (st andardmat rixfor T1 )
0 1
 1 0
0 1 (st andardmat rixfor T2 )
1 0
Elementary Linear Algebra: Section 6.3, p.392
54/78
T hestandardmatrixfor T  T2  T1
1  1 0 2 1 0 2 1 0
A  A2 A1  0 0 1 0 0 0  1 0 1


 

0
1
0
1
0
1
0
0
0


 

T hestandardmatrixforT '  T1  T2
2 1 0 1  1 0 2  2 1
A'  A1 A2  0 0 0 0 0 1  0 0 0
1 0 1 0 1 0 1 0 0
Elementary Linear Algebra: Section 6.3, p.392
55/78

Inverse linear transformation:
If T1 : Rn  Rn andT2 : Rn  Rn are L.T .s.t.for everyv inRn
T2 (T1 ( v))  v and T1 (T2 ( v))  v
T henT2 is called theinverseof T1 andT1 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 n  R n be a L.T .with standardmatrixA,
T hen thefollowingconditionare equivalent.
(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)
TheL.T.T： R3  R3 is definedby
T ( x1, x2 , x3 )  (2x1  3x2  x3 , 3x1  3x2  x3 , 2x1  4x2  x3 )
Show that T is invertible, and find its inverse.
Sol:
T hestandardmatrixfor T
2 3 1
A  3 3 1
2 4 1
 2 x1  3x2  x3
 3x1  3x2  x3
 2 x1  4 x2  x3
 2 3 1 1 0 0
 A I 3    3 3 1 0 1 0


2
4
1
0
0
1


Elementary Linear Algebra: Section 6.3, p.393
58/78
0
1 0 0  1 1
. J .E
G
0 1 0  1 0
1  I


0
0
1
6

2

3



A1

T hereforeT is invertibleand thestandardmatrixfor T 1 is A1
0
 1 1
A1   1 0
1


6

2

3


0   x1    x1  x2 
 1 1
T 1 ( v)  A1 v   1 0
1   x2     x1  x3 

  

6

2

3
x
6
x

2
x

3
x
2
3

 3   1
In other words,
T 1 ( x1 , x2 , x3 )  ( x1  x2 ,  x1  x3 , 6 x1  2 x2  3x3 )
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 , v2 ,, vn }
(a L.T ).
(a basis forV )
B'  {w1 , w2 ,, wm } (a basis forW )
Thus, the matrix of T relative to the bases B and B' is
A  T (v1 )B' , T (v2 )B' ,, T (vn )B'  M mn
Elementary Linear Algebra: Section 6.3, p.394
60/78

Transformation matrix for nonstandard bases:
Let V andW be finite- dimensional vectorspaces with basis B and B' ,
respectively,where B  {v1 , v2 ,, vn }
If T : V  W is a L.T. s.t.
T (v1 )B '
 a11 
 a12 
 a1n 
 a21 
 a22 
 a2 n 
  , T (v2 )B '   ,  , T (vn )B '   
  
  
  
am1 
am 2 
amn 
then them n matrix whose n columnscorrespondtoT (vi )B'
Elementary Linear Algebra: Section 6.3, p.395
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 a11
 a21
A  T (e1 ) T (e2 )  T (en )  
 
am1
a12
a22

am 2
 a1n 
 a2 n 

  
 amn 
is such thatT ( v)B'  A[ v]B for everyv in V .
Elementary Linear Algebra: Section 6.3, p.395
62/78

Ex 5: (Finding a matrix relative to nonstandard bases)
Let T： R2  R2 be a L.T .definedby
T ( x1 , x2 )  ( x1  x2 , 2 x1  x2 )
Find thematrixof T relativeto thebasis
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'  03, T (1, 1)B'  03
 
 
the matrix for T relative to B and B'
3 0 
A  T (1, 2)B ' T (1, 2)B '   
0  3
Elementary Linear Algebra: Section 6.3, p.395
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
Ex 6:
For theL.T .T： R 2  R 2 given in Example5, use thematrix A
to findT ( v), where v  (2, 1)
Sol:
v  (2, 1)  1(1, 2)  1(1, 1)
B  {(1, 2), (1, 1)}
1
 v B   
 1
3 0   1  3
 T ( v)B '  AvB  
 



0  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
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
Notes:
(1)In thespecialcase whereV  W and B  B' ,
thematrix A is alled thematrixof T relativeto thebasis B
(2)T : V  V : theidentity ransformat
t
ion
B  {v1 , v2 , , vn } : a basis forV
 thematrixof T relativeto thebasis B
1 0  0
0 1  0 
  In
A  T (v1 )B , T (v2 )B , , T (vn )B   
   


0
0

1


Elementary Linear Algebra, Section 6.3, p.396
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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
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6.4 Transition Matrices and Similarity
T :V  V
B  {v1 , v2 ,, vn }
( a L.T ).
( a basis of V )
B'  {w1 , w2 ,, wn } (a basis of V )
A  T (v1 )B , T (v2 )B ,, T (vn )B 
A'  T (w1 )B' , T (w2 )B' ,, T (wn )B' 
P  w1 B , w2 B ,, wn B 
P1  v1 B' , v2 B' ,, vn B' 
( matrixof T relativeto B)
(matrixof T relativeto B' )
( transition matrixfromB' to B )
( transition matrixfromB to B' )
vB  PvB' , vB'  P 1vB
T ( v)B  AvB
T ( v)B'  A' vB'
Elementary Linear Algebra: Section 6.4, p.399
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
Two ways to get from v B ' to T ( v)B ':
(1)(direct)
indirect
A'[ v]B '  [T ( v)]B '
(2)(indirect)
P 1 AP[ v]B '  [T ( v)]B '
1
 A'  P AP
Elementary Linear Algebra: Section 6.4, pp.399-400
direct
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
Ex 1: (Finding a matrix for a linear transformation)
Find thematrixA' for T： R 2  R 2
T ( x1 , x2 )  (2x1  2 x2 ,  x1  3x2 )
reletiveto thebasis B'  {(1, 0), (1, 1)}
Sol:
(I) A'  T (1, 0)B'
T (1, 1)B' 
3
T (1, 0)  (2,  1)  3(1, 0)  1(1, 1)  T (1, 0)B '   
 1
 2
T (1, 1)  (0, 2)  2(1, 0)  2(1, 1)  T (1, 1)B '   
2
 3  2
 A'  T (1, 0)B ' T (1, 1)B '   
 1 2 
Elementary Linear Algebra: Section 6.4, p.400
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(II)standardmatrixfor T ( matrixof T relativeto B  {(1, 0), (0, 1)})
 2  2
A  T (1, 0) T (0, 1)  
 1 3 
transitionmatrixfrom B' to B
1 1
P  (1, 0)B (1, 1)B   

0
1


transitionmatrixfrom B to B'
1  1
P 

0
1


matrixof T relativeB'
1
1  1  2  2 1 1  3  2
A'  P AP  







0
1

1
3
0
1

1
2



 

1
Elementary Linear Algebra: Section 6.4, p.400
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
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 7 
2
2
and let A  
be
the
matrix
for
T
:
R

R
relativeto B.

  3 7
Find thematrixof T relativeto B'.
Sol:
3  2
transitionmatrixfrom B' to B : P  (1, 2)B (2,  2)B   

2

1


  1 2
1
transitionmatrixfrom B to B': P  (3, 2)B ' (4,  2)B '   


2
3


matrix of T relative to B':
  1 2  2 7 3  2  2 1
A'  P AP  







 2 3   3 7 2  1  1 3
1
Elementary Linear Algebra: Section 6.4, p.401
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
Ex 3: (Finding a matrix for a linear transformation)
For thelinear transformation T : R 2  R 2 given in Ex.2,find v B、
T ( v)B and T ( v)B ' , for thevectorv whose coordinatematrixis
 3
vB '   
  1
Sol:
3  2  3  7
vB  PvB'  2  1   1   5

   
 2 7  7  21
T ( v)B  AvB    3 7  5   14

  

  1 2  21  7
1
T ( v)B'  P T ( v)B   2 3  14   0 


  
or T ( v)B '  A' vB '
 2 1  3  7

 



 1 3   1  0 
Elementary Linear Algebra: Section 6.4, p.401
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

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 AIn
(2) A  P 1 BP  PAP1  P( P 1BP) P 1
PAP1  B  Q 1 AQ  B (Q  P 1 )
Elementary Linear Algebra: Section 6.4, p.402
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
Ex 4: (Similar matrices)
 2  2
 3  2
(a) A  
and A'  
are similar


 1 3 
 1 2 
1 1
1
because A'  P AP, where P  

0
1


 2 7 
 2 1
(b) A  
and A'  
are similar


  3 7
 1 3
3  2
because A'  P AP, where P  

2

1


1
Elementary Linear Algebra: Section 6.4, p.403
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
Ex 5: (A comparison of two matrices for a linear transformation)
1 3 0 
Suppose A  3 1 0  is thematrixfor T : R 3  R 3 relative
0 0  2
to thestandardbasis. Find thematrixfor T relativeto thebasis
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
(1,  1, 0)B
1
 12
2
1
1
 P   2  12

0 0
0
0

1
Elementary Linear Algebra: Section 6.4, p.403
1 1 0
(0, 0, 1)B   1  1 0
0 0 1
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matrixof T relativeto B':
 12 12
A'  P 1 AP   12  12
0 0
0
4 0
 0  2 0 
0 0  2
Elementary Linear Algebra: Section 6.4, p.403
0 1 3 0  1 1 0
0 3 1 0  1  1 0
1 0 0  2 0 0 1
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
Notes: Computational advantages of diagonal matrices:
d1k
0
k
(1) D  

 0
0
d 2k

0
 0
 0

  
 d nk 
0




 d1
0
D

 0
0
d2

0
 0
 0

 
 d n 
(2) DT  D
 d11
0
(3) D 1  

0

1
d2

0
Elementary Linear Algebra: Section 6.4, p.404
0
0
, d i  0

1 
dn 
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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