Page 626 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

To focus on the latter idea, let be a linear transformation. As shown in Figure 8.4.5, the matrix can be used

to calculate in three steps by the following indirect procedure:

1. Compute the coordinate vector .

Figure 8.4.5

2. Multiply on the left by to produce .

3. Reconstruct from its coordinate vector .

EXAMPLE 6 Linear Operator on .
Let be the linear operator defined by .
that is,

(a) Find with respect to the basis

(b) Use the indirect procedure to compute .
(c) Check the result in (b) by computing directly.

Solution (a)

From the formula for T,

so

Thus