Chapter 4
Linear Transformations
1 Definition and Examples
Definition
A mapping L from a vector space V into a vector space W is
said to be a linear transformation if
(1)
L( α v1+βv2)= α L(v1)+βL(v2)
for all v1, v2 ∈V and for all scalars α and β.
Linear Operators on R2
Example
Let L be the operator defined by L(x)=3x
for each x∈R2.
Example
Consider the mapping L defined by L(x)=x1e1
for each x∈R2.
Example
Let L be the operator defined by L(x)=(x1,-x2)T
for each x=(x1,x2)T∈R2.
Linear Transformations from Rn to Rm
Example
The mapping L: R2→R1 defined by L(x)=x1+x2
is a linear transformation.
Example
Consider the mapping M defined by
M (x)  ( x  x )
2
1
Example
2
2
1
2
The mapping L from R2 to R3 defined by
L(x)=(x2 , x1, x1+x2)T is a linear transformation.
Linear Transformations from V to W
If L is a linear transformation mapping a vector space V into
a vector space W, then
(1) L(0v)=0w (where 0v and 0w are the zero vectors in V and W,
respectively).
(2) If v1, …, vn are elements of V and α1, …, αn are scalars, then
L(α1v1+ α2v2+‥‥+ αnvn)= α1L(v1)+ α2L(v2）+‥‥+ αnL(vn)
(3) L(-v)=-L(v) for all v∈V.
The Image and Kernel
Definition
Let L: V→W be a linear transformation. The kernel of L,
denoted Ker(L), is defined by
Ker(L)={ v∈V ∣L(v)=0w}
Definition
Let L: V→W be a linear transformation and let S be a
subspace of V. The image of S, denoted L(S), is defined by
L(S)={ w∈W ∣w=L(v) for some v∈ S}
The image of the entire vector space, L(V), is called the
range of L.
Theorem 4.1.1 If L: V →W is a linear transformation
and S is a subspace of V, then
(1) Ker(L) is a subspace of V.
(2) L(S) is a subspace of W.
Example
Let L be the linear operator on R2 defined by
 x1 
L( x )   
0
Example
Let L: R3→R2 be the linear transformation
defined by L(x)=(x1+x2, x2+x3)T and let S be the subspace
of R3 spanned by e1 and e3.
2 Matrix Representations of Linear
Transformation
Theorem 4.2.1 If L is a linear transformation mapping Rn
into Rm, there is an m×n matrix A such that
L(x)=Ax
for each x∈Rn. In fact, the jth column vector of A is given
by
Example
aj=L(ej)
j=1, 2, …, n
Define the linear transformation L: R3→R2 by
L(x)=(x1+x2, x2+x3)T
Theorem 4.2.2 (Matrix Representation Theorem)
If E=[v1, v2, …,vn] and F=[w1,w2, … ,wm] are ordered bases
for vector spaces V and W, respectively, then corresponding
to each linear transformation L: V → W there is an m×n
matrix A such that
[L(v)]F=A[v]E
for each v∈V
A is the matrix representing L relative to the ordered bases
E and F. In fact,
aj=[L(vj)]F
j=1, 2, …, n
If A is the matrix representing L with respect to the bases E
and F and
x=[v]E
(the coordinate vector of v with respect to E)
y=[w]F (the coordinate vector of w with respect to F)
then L maps v into w if and only if A maps x into y.
v∈V
x=[v]E
∈Rn
L=LA
A
w=L(v) ∈W
Ax=[w]F∈Rm
Example
Let L be a linear transformation mapping R3
into R2 defined by L(x)=x1b1+(x2+x3)b2 for each x∈R3,where
 1
b1   
 1
and
  1
b 2   
1
Find the matrix A representing L with respect to the ordered
bases [e1, e2, e3] and [b1, b2].
Theorem 4.2.3 Let E=[u1, u2, …,un] and F=[b1,b2, … ,bm]
be ordered bases for Rn and Rm, respectively. If L: Rn → Rm
is a linear transformation and A is the matrix representing
L with respect to E and F, then
aj=B-1L(uj)
where B=(b1, …,bm).
for j=1, 2, …, n
Corollary 4.2.4 If A is the matrix representing the linear
transformation L: Rn→Rm with respect to the bases
E=[u1, u2, …,un] and F=[b1,b2, … ,bm]
then the reduced row echelon form of (b1, …,bm ∣L(u1), …,L(un))
is (I∣A).
Example
Let L: R2→R3 be the linear transformation
defined by
L(x)=(x2, x1+x2, x1-x2)T
Find the matrix representations of L with respect to the ordered
bases [u1, u2] and [b1, b2, b3], where
u1=(1, 2)T,
u2=(3, 1)T
and
b1=(1, 0, 0)T, b2=(1, 1, 0)T,
b3=(1, 1, 1)T
2 Similarity
If L is a linear operator on an n-dimensional vector space V,
the matrix representation of L will depend on the ordered
basis chosen for V. By using different bases, it is possible
to represent L by different n×n matrices.
Example
Let L: R2→R2 be the linear transformation
defined by
L(x)=(2x1, x1+x2)T

So the matrix representing L with respect to [e1, e2] is A  

1 1
2 0
If there is another basis for
R2:
 2 0 1  2 
L(u1)=Au1= 
    
 1 1 1  2 
1
  1
u1   ， u 2   
1
1
 2 0   1   2 
L(u2)=Au2= 
    
 1 1  1   2 
1  1
The transition matrix from [u1, u2] to [e1, e2] is U=(u1, u2)= 

1 1 
 1

The transition matrix from [e1, e2] to [u1, u2] is U-1=  2
  1
 2
Let matrix B representing L with repect to [u1, u2],
2
b1=U-1L(u1)=  
 
0
1
b2=U-1L(u2)=  
1
 
 2  1

B  
0 1 
1

2
1

2
Thus, if
(1) B is the matrix representing L with respect to [u1, u2]
(2) A is the matrix representing L with respect to [e1, e2]
(3) U is the transition matrix corresponding to the change of
basis from [u1, u2] to [e1, e2]
then
B=U-1AU
Theorem 4.3.1 Let E=[v1, v2, …,vn] and F=[w1,w2, … ,wn] be
two ordered bases for a vector space V, and let L be a linear
operator on V. 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-1AS.
Definition
Let A and B be n×n matrices. B is said to be similar to A if
there exists a nonsingular matrix S such that B=S-1AS.
Example
Let L be the linear operator on R3 defined by
L(x)=Ax, where
 2 2 0


A   1 1 2
 1 1 2


Thus the matrix A representing L with respect to [e1, e2, e3] .
Find the matrix representing L with respect to [y1, y2, y3], where
1
  2
1
 
 
 
y1    1, y 2   1 , y3  1
0
 1 
1
 
 
 
Some Properties for Similar Matrices
1、Matrix A is similar to A itself.
2、If B is similar to A, then A is also similar to B.
3、If A is similar to B and B is similar to C, then A is similar
to C.
4、If A and B are similar matrices, then det(A)=det(B).
5、If A and B are similar matrices, then AT is similar to BT
6、If A and B are similar matrices, then Ak and Bk are similar
for each positive integer k.