Page 588 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

and

Thus satisfies the two requirements of a linear transformation.

EXAMPLE 15 Composition of Linear Transformations
Let and be the linear transformations given by the formulas

Then the composition is given by the formula

In particular, if , then

EXAMPLE 16 Composition with the Identity Operator

If is any linear operator, and if is the identity operator (Example 3), then for all vectors v in V , we
have

It follows that and are the same as T ; that is,

(3)

We conclude this section by noting that compositions can be defined for more than two linear transformations. For example,
if

are linear transformations, then the composition is defined by
(Figure 8.1.6).
(4)