Page 84 - Advanced Linear Algebra

68 Advanced Linear Algebra

is a standard basis vector, we conclude that
and since ´ µ ~ 8

´ b ! µ 89Á ~ ´ µ 89Á b !´ µ 89Á
and so is linear. If ( C Á , we define by the condition ´ µ ~ ( ² ³ ,

9

´
²
whence ³ ~( and is surjective. Also, ker ² ³ ~¸ ¹ since µ ~
8
)

implies that ~ . Hence, the map is an isomorphism. To prove part 2 , we

have

8
´ 8:Á µ ´#µ ~ ´² #³µ ~ ´µ 9 :, ´ #µ ~ ´µ 9 :Á 9 ´ µ 89Á ´#µ 8
:
Change of Bases for Linear Transformations

8
9
Since the matrix ´µ 89, that represents depends on the ordered bases and , it
is natural to wonder how to choose these bases in order to make this matrix as
simple as possible. For instance, can we always choose the bases so that is

represented by a diagonal matrix?
As we will see in Chapter 7, the answer to this question is no. In that chapter,
we will take up the general question of how best to represent a linear operator
by a matrix. For now, let us take the first step and describe the relationship

between the matrices ´µ 89Á and ´µ 8 9Á Z Z of with respect to two different pairs
Z
²Á ³ and ² Á ³ of ordered bases. Multiplication by ´ µ Á89 Z sends ´#µ to
Z
89

9
8
Z
Z
8
´#µ 9 . This can be reproduced by first switching from 8 Z to , then applying

8
Z
´µ 89Á and finally switching from to , that is,
Z

9
9
c

´µ 8 9, Z Z ~ 4 99Á Z ´µ 8 9, 4 8 8Á Z ~ 4 99Á Z ´ µ 8 9, 4 88 Z
Á
Z

Z
9
9
B

Theorem 2.16 Let ²= ,> ³ and let 8 ² Á ³ and 8 ² Á ³ be pairs of ordered
bases of and > , respectively. Then
=
(2.1)
´µ 89Á Z Z ~ 4 9 9Á Z ´µ 8 9Á 4 88Á Z
When B ²= ³ is a linear operator on , it is generally more convenient to
=
represent by matrices of the form ´µ 8 , where the ordered bases used to
8
represent vectors in the domain and image are the same. When ~ 9 , Theorem
2.16 takes the following important form.
=
B
8
Corollary 2.17 Let ²= ³ and let and be ordered bases for . Then the

9
matrix of with respect to can be expressed in terms of the matrix of with

9
respect to as follows:
8
c

9
8 Á
´µ ~ 4 8 9 ´µ 4 89 (2.2)
Á
Equivalence of Matrices
Since the change of basis matrices are precisely the invertible matrices, 2.1 has
(
)
the form