Chapter 4---Section 1 Definitions and Examples
Chapter 4
Linear Transformations
In this chapter, we introduce the general concept of linear
transformation from a vector space into a vector space. But, we mainly
focus on linear transformations from Rn
to R m .
§1 Definition and Examples
New words and phrases
Mapping 映射
Linear transformation 线性变换
Linear operator 线性算子
Dilation 扩张
Contraction 收缩
Projection 投影
Reflection 反射
Counterclockwise direction 反时针方向
Clockwise direction 顺时针方向
Image 像
Kernel 核
1.1 Definition
★Definition A mapping (映射) L: VW is a rule that produces a
correspondence between two sets of elements such that to each element in
the first set there corresponds one and only one element in the second set.
★Definition A mapping L from a vector space V into a vector space
W is said to be a linear transformation （线性变换）if
Chapter 4---Section 1 Definitions and Examples
(1)
L(1v1   2 v2 )  1L(v1 )   2 L(v2 )
for all v1 , v2 V and for all scalars 1 and  2 .
(1) is equivalent to
(2)
L(v1  v2 )  L(v1 )  L(v 2 ) for any v1 , v2 V
(3)
L( v)   L(v) for any v V and scalar  .
and
Notation: A mapping L from a vector space V into a vector space W is
denoted
L: VW
When W and V are the same vector space, we will refer to a linear
transformation L: VV as a linear operator on V. Thus a linear operator
is a linear transformation that maps a vector space V into itself.
1.2 Linear Operators on
R2
1. Dilations(扩张) and Contractions
Let L be the operator defined by
L(x)=kx
then this is a linear operator. If k is a positive scalar, then the linear
operator can be thought of as a stretching or shrinking by a factor of k.
 k 0   x1 
L(x)  
    Ax
 0 k  x2 
2. Projection （投影）onto the coordinate axes.
Chapter 4---Section 1 Definitions and Examples
L(x)= x1e1
 1 0   x1 
L(x)  
    Ax
 0 0  x2 
L(x)= x2 e 2
 0 0   x1 
L(x)  
    Ax
 0 1  x2 
3. Reflections (反射) about an axis
Let L be the operator defined by
L(x)= ( x1 ,  x2 )T , then it is a linear operator. The operator L has the
 1 0   x1 
    Ax
 0 1 x2 
effect of reflecting vectors about the x-axis. L(x)  
Reflecting about the y-axis
L(x)= ( x1 , x2 )T ,
 1 0   x1 
L(x)  
    Ax
 0 1  x2 
4. Rotations
L(x)= ( x2 , x1 )T , L has the effect of rotating each vector by 90 degrees
in the counterclockwise direction（逆时针方向）.
 0 1  x1 
L(x)  
    Ax
 1 0  x2 
1.3 Linear Transformations from
Rn
to
Rm
If A is an mxn matrix, then we can define a linear transformation LA
from Rn to R m by
LA ( X )  AX
It is easy to verify that the mapping above is linear. In the next
section, we will see that any linear transformation from Rn to R m
must be of this form.
Chapter 4---Section 1 Definitions and Examples
1.4 The Image and Kernel
★ Definition
Let L: Rn  R m is a linear transformation. The
kernel （核）of L denoted ker(L), is defined by
ker(L)= v V | L(v)  0W 
★Definition Let L: Rn  R m is a linear transformation and let S
be a subspace of V. The image （像）of S, denoted L(S), is defined by
L(S)= w  R m | w  L(v) for some v  R n 
The image of the entire vector space, L(V), is called the range（值域）of
L.
Theorem 4.1.1 If L: Rn  R m is a linear transformation and S is a
subspace of Rn , then
(i)
ker(L) is a subspace of Rn .
(ii)
L(S) is a subspace of R m .
Assignment for section 1, chapter 4
Hand in: 3, 4, 17, 20,
Not required : 8, 10, 11, 15, 16, 19, 25
Chapter 4---Section 2 Matrix Representations
§2 Matrix Representations of Linear Transformations
New words and phrases
Matrix representation 矩阵表示
Formal multiplication 形式乘法
Similarity 相似性
2.1 Matrix Representation of Linear Transformations
In section 1 of this chapter, the examples of linear transformations
can be represented by matrices. In general, a linear transformation can
be represented by a matrix.
If we use the basis E=[ u1 , u 2 ,
F=[ v1 , v2 ,
, vm ] for V, and
, u n ] for U and the basis
L: U  V.
If u is a vector in U, then
u  x1u1  x2 u 2 
 xn u n (in U) | L(u)  y1v1  y2 v2 
 ym vm (in V)
The linear transformation L is determined by the change of the
coordinate vectors:
 x1   y1 
   
 x 2    y2 
   
   
 x n   ym 
Assume that
L(u j )  a1 j v1  a2 j v 2 
Formally,
 amj v m , j=1, 2, …, n
Chapter 4---Section 2 Matrix Representations
, L(u n )] = [v1 , v2 ,
[ L(u1 ), L(u 2 ),
 a11

a
, vm ]  21


 am1
a12
a22
am 2
a1n 

a2 n 


amn 
Write [ L(u1 ), L(u 2 ), , L(u n )] as linear combinations of [v1 , v2 , , vm ] , then
consequently, A is obtained.
Then L(u) = L( x1u1  x2 u 2   xn u n )
= [v1 , v2 ,
 y1 
 
y
, vm ]  2 
 
 
 ym 
(formal
multiplication)
 x1L(u1 )  x2 L(u 2 ) 
n
 xn L(u n ) =  x j L(u j )
j=1
 [L(u1 ), L(u 2 ),
n
m
j =1
i 1
 x1 
 
x
, L(u n )]  2  (formal multiplication)
 
 
 xn 
=  x j ( aij vi )
m
n
i=1
j 1
=  ( aij x j )vi = [v1 , v2 ,
 a11

a
, vm ]  21


 am1
a12
a22
am 2
a1n   x1 
 
a2 n   x2 
 
 
amn   xn 
Hence
 y1   a11
  
 y2  =  a21
  
  
 ym   am1
a12
a22
am 2
a1n   x1 
 
a2 n   x2 
 
 
amn   xn 
Thus, y=Ax is the coordinate vector of L(u) with respect to
F=[ v1 , v2 ,
, vm ]. y=Ax is called the matrix representation of the
linear transformation. A is called the matrix representing L relative to the
Chapter 4---Section 2 Matrix Representations
bases E and F. A is determined by the following equations.
[ L(u1 ), L(u 2 ),
, L(u n )] = [v1 , v2 ,
 a11

a
, vm ]  21


 am1
a12
a22
am 2
a1n 

a2 n 


amn 
We have established the following theorem.
Theorem 4.2.2 If E=[ u1 , u 2 ,
F=[ v1 , v2 ,
, u n ] is an ordered basis for U and
, vm ] is an ordered basis for V, then corresponding to each
linear transformation L:UV there is an mxn matrix A such that
[ L(u )]F  A[u ]E for each u in U.
A is the matrix representing L relative to the ordered bases E and F.
In fact, a j  [ L(u j )]F .
2.2 Matrix Representation of L:
Rn  R m
If U= Rn , V= R m , then we have the following theorem.
Theorem 4.2.1 If L is a linear transformation mapping Rn into R m ,
there is an mxn matrix A such that
L(x)=Ax
for each x  Rn . In fact, the jth column vector of A is given by
A  ( L(e1 ), L(e2 ),
Proof
, L(en ))
If we choose standard basis [e1 , e2 , , en ] for Rn and the
Chapter 4---Section 2 Matrix Representations
standard basis [e1 , e 2 , , e m ] for R m ,
 y1   y1 
   
y
y
, em )  2  =  2 
   
   
 ym   ym 
L(x)= L( x1e1  x2e2   xnen ) = (e1 , e 2 ,
 x1L ( e1 ) x
2
n
L (2e ) xn
n
=
L
( e x j)L(e j )
j=1
 (L(e1 ), L(e 2 ),
 x1 
 
x
, L(e n ))  2 
 
 
 xn 
And let A=  aij  =  a1 , a 2 ,
, a n   (L(e1 ), L(e2 ),
, L(en ))
If x  x1e1  x2e2   xnen , then L(x)=Ax.
A is referred to as the standard matrix representation(标准矩阵表示)
of L.( A representation with respect to the standard basis.)
Example 1 (example 1 on page 186) Determine the standard matrix
representation of L.
Define the linear transformation L: R3  R2 by
L(x)  ( x1  x2 , x2  x3 )T for each x  ( x1 , x2 , x3 )T in R 3 , find the linear
standard representation of L.
1 1 0

0 1 1
Solution: Find L(e1 ), L(e2 ), L(e3 ) . Then A  (L(e1 ), L(e2 ), L(e3 ))  
Example 2 rotation by an angle 
Let L be the linear transformation operator on R2 that rotates each
vector by an angle  in the counterclockwise direction. We can see that
Chapter 4---Section 2 Matrix Representations
e1 is mapped to (cos  ,sin  )T , and e2 is mapped to ( sin  ,cos  )T .
L(e1 )  (cos  ,sin  )T , L(e 2 )  (sin  ,  cos  )T
The matrix A representing the transformation will be
 cos 
A  (L(e1 ), L(e2 ))  
 sin 
 sin  

cos  
To find the matrix representation A for a linear transformation L Rn
 R m w.r.t. the bases E=[ u1 , u 2 ,
, u n ] and F=[ b1 , b2 ,
must represent each vector L(u j )  a1 j b1  a2 j b 2 
, b m ], we
 amj b m . The following
theorem shows that determining this representation is equivalent to
solving the linear system Bx= L(u j ) , where L(u j ) is regarded as a
column vector in R m .
Theorem 4.2.3 Let E=[ u1 , u 2 ,
, u n ] and F=[ b1 , b2 ,
, b m ] be
ordered bases for Rn and R m , respectively. If L: Rn  R m is a linear
transformation and A is the matrix representing L with respect to E and F,
then
A  B 1 ( L(u1 ), L(u 2 ),
where B=( b1 , b2 ,
, L(u n ))
, b m ).
Proof L(u) = L( x1u1  x2 u 2   xn u n ) = (b1 , b2 ,
 x1L(u1 )  x2 L(u 2 ) 
n
 y1 
 
y
, bm )  2 
 
 
 ym 
 xn L(u n ) =  x j L(u j )
j=1
Chapter 4---Section 2 Matrix Representations
 x1 
 
x
, L(u n ))  2 
 
 
 xn 
 (L(u1 ), L(u 2 ),
 y1 
 
y
, bm )  2   (L(u1 ), L(u 2 ),
 
 
 ym 
(b1 , b2 ,
 x1 
 
x
, L(u n ))  2 
 
 
 xn 
The matrix B is nonsingular since its column vectors form a basis for
R m . Hence, A  B 1 ( L(u1 ), L(u 2 ),
( L(u1 ), L(u2 ),
[u1 , u 2 ,
, L(u n ))
[e1 ,e2 ,
,em ]
B
B1
[b1 , b2 ,
, bm ]
, L(un ))
, un ]
A
( L(u1 ), L(u 2 ),
[ u1 , u 2 ,
, L(u n )) is the matrix representing L relative to the bases
, u n ] and [ e1 , e2 ,
, em ]. B is the transition matrix
corresponding to the change of basis from [ b1 , b2 ,
[ e1 , e2 ,
, b m ] to
, em ].
Corollary 4.2.4 If A is the matrix representing the linear transformation L:
Rn  R m with respect to the bases
Chapter 4---Section 2 Matrix Representations
E=[ u1 , u 2 ,
, u n ] and F=[ b1 , b2 ,
, bm ]
then the reduced row echelon form of
(b1 , b2 ,
, bm | L(u1 ), L(u 2 ),
, L(u n ))
is (I|A)
Proof (b1 , b2 ,
, bm | L(u1 ), L(u 2 ),
, L(u n )) =(B|BA), which is row
equivalent to (I|A).
Examples Finding the matrix representing L
Example 3 on page 188
Let L be a linear transformation mapping R3 into R2 defined by
L ( x) x1
1
the matrix A representing L with respect to
b x 2(  x 3 . )Find
2b
1
1
 1 

1
the ordered bases [e1 , e2 , e3 ] and [b1 , b2 ] , where b1    , b 2  
Solution:
Method 1. Represent [e1 , e2 , e3 ] in terms of [b1 , b2 ]
Method 2. A  (b1 ,b 2 )1 ( L(e1 ), L(e 2 ), L(e3 ))
1
1 1 1 1 1  1/ 2 1/ 2 1 1 1  1 0 0 
A
 




1 1  1 1 1   1/ 2 1/ 2 1 1 1   0 1 1 
Method 3 Applying row operations.
(b1 ,b2 | L(e1 ), L(e2 ), L(e3 ))
Example 4 on page 188
Let L be a linear transformation mapping R2 into itself defined by
L ( 1b  2b ) (  1) b  2,2 where
b
[b1 , b 2 ] is the ordered basis
Chapter 4---Section 2 Matrix Representations
defined in example 3. Find the matrix A representing L with respect to
[b1 , b 2 ] .
Solution: Use three methods as in example 3.
Example 6 on page 190
Determine the matrix representation of L with respect to the given bases.
Let L: R2  R3 be the linear transformation defined by
L ( x) x2( x1 , x2 x1, xT 2 )
Find the matrix representation of L with respect to the ordered bases
[u1 , u 2 ] and [b1 , b2 , b3 ] , where
u1  (1, 2)T , u 2  (3,1)T
b1  (1, 0, 0)T , b 2  (1,1, 0)T , b3  (1,1,1)T
Assignment for section 2, chapter 4
Hand in: 2, 6, 8, 16, 20
Not required: 9—15, 17, 19
Chapter 4---Section 3 Similarity
§3 Similarity
Let L be a linear operator on V, E=[ v1 , v2 ,
, vn ] be an ordered
basis for V, A is the matrix representing L with respect to the basis E.
u  x1 v1 x 2 v2
L(v j )  a1 j v1  a2 j v2 
 xn nv, L(u)  y1v1  y2 v2 
 anj vn  [v1 , v 2 ,
 yn v n
 a1 j 
 
a2 j
, v n ]    [v1 , v 2 ,
 
 
 anj 
, v n ]a j
y=Ax
F=[ w1 , w 2 ,
, wn ]
u  c1w1  c2 w 2 
 cn w n , L(u)  d1w1  d2 w 2 
L(w j )  b1 j w1  b2 j w 2 
 bnj w n  [w1 , w 2 ,
 dn w n
 b1 j 
 
b2 j
, w n ]    [w1 , w 2 ,
 
 
 bnj 
, w n ]b j
d=Bc
Let the transition matrix corresponding the change of basis from
F=[ w1 , w 2 ,
, w n ] to [ v1 , v2 ,
Then x=Sc, y=Sd, S 1 y  BS 1x
, vn ]
y  SBS 1x or A  SBS 1
Chapter 4---Section 3 Similarity
Coordinate vector of u: x
Basis E= [v1 , v2 , , vn ] V
uL(u)
V
Coordinate vector of L(u) :y
Basis E
Ax=y
x=Sc
S-1
S
Coordinate vector of u: c
Basis E= [w1 , w 2 ,
, wn ] V
y=Sd
uL(u)
Bc=d
V Coordinate vector of L(u): d
Basis F
Hence, we have established the following theorem.
Theorem 4.3.1 Let E=[ v1 , v2 ,
, vn ] and F=[ w1 , w 2 ,
, w n ] be
two ordered bases for a vector space V, and let L be a linear operator on
Rn . Let S be the transition matrix representing the change from F to E. If
A is the matrix representing L with respect to E, and B is the matrix
representing L with respect to F, then B  S 1 AS .
★Definition Let A and B be nxn matrices. B is said to be similar to
A if there is a nonsingular matrix S such that B  S 1 AS .
Example 2 (on page 204)
Example Let L be the linear operator on R3 defined by L(x)=Ax, where
Chapter 4---Section 3 Similarity
2 2 0


1 1 2 .
1 1 2


Thus the matrix A represents L with respect to the standard basis for R3 .
Find the matrix representing L with respect to the basis [ y1 , y2 ,y3 ], where
y1  ( 1, 1T, ,0 y) 2  ( 2,1,1)T , y3  (1,1,1)T .
Solution
0 0 0
D=  0 1 0  is the matrix representing L w.r.t the basis
0 0 4


[ y1 , y2 ,y3 ],.
Or, we can find D using D  Y 1 AY
An x  (YDY 1 )n x=(YDnY 1 )x
Using this example to show that it is desirable to find as simple as a
representation as possible for a linear operator. In particular, if the
operator can be represented by a diagonal matrix, this is usually preferred
representation. It makes the computation of Dx and D n x easier.
Assignment for section 3, chapter 4
Hand in: 2, 3, 4, 8, 10, 15
Not required 5, 6