Page 628 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a) T is one-to-one.
(b) is invertible.
Moreover, when these equivalent conditions hold,

(14)

Remark In 13, observe how the interior subscript (the basis for the intermediate space V ) seems to “cancel out,” leaving
only the bases for the domain and image space of the composition as subscripts (Figure 8.4.6). This cancellation of interior
subscripts suggests the following extension of Formula 13 to compositions of three linear transformations (Figure 8.4.7).

Figure 8.4.6

Figure 8.4.7 (15)
The following example illustrates Theorem 8.4.2.

EXAMPLE 7 Using Theorem 8.4.2
Let be the linear transformation defined by

and let be the linear operator defined by

Then the composition is given by

Thus, if , then

In this example, plays the role of U in Theorem 8.4.2, and plays the roles of both V and W; thus we can take (16)
13 so that the formula simplifies to in