Page 85 - Advanced Linear Algebra

Linear Transformations 69

´µ 89Á Z Z ~ 7´µ 8 9Á 8 c
where and are invertible matrices. This motivates the following definition.
7
8
Definition Two matrices ( and ) are equivalent if there exist invertible
matrices and for which
8
7
)~ 7(8 c
We have remarked that is equivalent to if and only if can be obtained
)
(
)
from by a series of elementary row and column operations. Performing the
(
row operations is equivalent to multiplying the matrix on the left by and
7
(
performing the column operations is equivalent to multiplying on the right by
(
c
8 .
(
In terms of 2.1 , we see that performing row operations premultiplying by )
)
(
7
is equivalent to changing the basis used to represent vectors in the image and
(
performing column operations postmultiplying by 8 c ) is equivalent to
changing the basis used to represent vectors in the domain.
(
According to Theorem 2.16, if and ) are matrices that represent with

respect to possibly different ordered bases, then and are equivalent. The
)
(
converse of this also holds.

Theorem 2.18 Let = and > be vector spaces with dim ² = ³ ~ and
dim²> ³ ~ . Then two d matrices ( and ) are equivalent if and only if

they represent the same linear transformation ²= Á > ³ , but possibly with
B
respect to different ordered bases. In this case, and represent exactly the
(
)
same set of linear transformations in B²= Á > ³ .
Proof. If and represent , that is, if

)
(

( ~ ´µ 89, and ) ~ ´µ 8 9, Z Z
Z
ÁÁ
9
for ordered bases 89 8 Z and , then Theorem 2.16 shows that and are
(
)
equivalent. Now suppose that and are equivalent, say
)
(
)~ 7(8 c
where 7 and 8 are invertible. Suppose also that ( represents a linear
9
transformation ²= Á > ³ for some ordered bases and , that is,

8
B

(~ ´ µ 89Á
=
Theorem 2.9 implies that there is a unique ordered basis 8 Z for for which
8~4 88Á and a unique ordered basis for 9 Z > for which 7 ~4 99Á . Hence
Z
Z
)~ 4 9 9Á Z ´ µ 8 9Á 4 88Á Z ~ ´ µ 89Á Z Z