Page 83 - Advanced Linear Algebra

Linear Transformations 67

vy vy vy

´+² ³µ~ ´ µ~ ´+²%³µ~ ´ µ~ , Á ´+²% ³µ~ ´ %µ~
9
9
9
9
9
9
wz wz wz

and so
v y
´+µ ~
8
w z
Hence, for example, if ²%³ ~ b%b % , then

v yvy vy
´+ ²%³µ ~ ´+µ ´ ²%³µ ~ ~
9
8
8
w zwz wz

and so + ²%³ ~ b % .
The following result shows that we may work equally well with linear
transformations or with the matrices that represent them with respect to fixed
(
ordered bases 8 and 9 ) . This applies not only to addition and scalar
multiplication, but also to matrix multiplication.

Theorem 2.15 Let and > be finite-dimensional vector spaces over , with
-
=
ordered bases ~² Á Ã Á ³ and ~² Á Ã Á ³ , respectively.
8
9

)
1 The map B¢²= Á > ³ ¦ C Á ²-³ defined by
²³ ~ ´ µ 89,
is an isomorphism and so B C²= Á > ³ š Á ²-³ . Hence,
dim² ²= Á > ³³~ dim² C Á ²-³³~ d
B
)
:
8
9

B

B
2 If ²<Á = ³ and ²= Á > ³ and if , and are ordered bases for
,
<= and >, respectively, then
´ µ 8:, ~ ´ µ 9 :, ´ µ 89,
)
(
Thus, the matrix of the product composition is the product of the
matrices of and . In fact, this is the primary motivation for the definition

of matrix multiplication.
Proof. To see that is linear, observe that for all ,

´ b ! µ 89 Á ´ µ ~ ´² b ! ³² ³µ 9
8

~´ ² ³ b ! ² ³µ 9

~ ´ ² ³µ b !´ ² ³µ 9 9
~ ´ µ 89 Á ´ µ b !´ µ 89 Á ´ µ 8
8

~² ´ µ 89 Á b !´ µ 89 Á ³´ µ 8