Page 639 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 1 Using Theorem 8.5.2
Let be defined by

Find the matrix of T with respect to the standard basis for ; then use Theorem 8.5.2 to find the matrix of T with

respect to the basis , where

Solution

We showed earlier in this section [see 2] that

To find from 10, we will need to find the transition matrix

[see 5]. By inspection,

so
Thus the transition matrix from to B is

The reader can check that
so by Theorem 8.5.2, the matrix of T relative to the basis is

which agrees with 4.

Similarity

The relationship in Formula 10 is of such importance that there is some terminology associated with it.

DEFINITION .
If A and B are square matrices, we say that B is similar to A if there is an invertible matrix P such that