Page 627 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution (b) is

The coordinate vector relative to B for the vector

Thus, from 5a,

from which it follows that

Solution (c)

By direct computation,

which agrees with the result in (b).

Matrices of Compositions and Inverse Transformations

We shall now mention two theorems that are generalizations of Formula 21 of Section 4.2 and Formula 1 of Section 4.3. The
proofs are omitted.

THEOREM 8.4.2

If and are linear transformations, and if B, , and are bases for U, V, and W, respectively,
then (13)

THEOREM 8.4.3
If is a linear operator, and if B is a basis for V, then the following are equivalent.