Page 514 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a) Find the transition matrix from to B.

(b) Use 5 to find if

Solution (a)

First we must find the coordinate vectors for the new basis vectors and relative to the old basis B. By inspection,

so

Thus the transition matrix from to B is

Solution (b)

Using 5 and the transition matrix in part (a) yields

As a check, we should be able to recover the vector v either from or . We leave it for the reader to show that
.

EXAMPLE 2 A Different Viewpoint on Example 1

Consider the vectors , , , . In Example 1 we found the transition matrix from the

basis for to the basis . However, we can just as well ask for the transition matrix from B to .

To obtain this matrix, we simply change our point of view and regard as the old basis and B as the new basis. As usual, the

columns of the transition matrix will be the coordinates of the new basis vectors relative to the old basis.

By equating corresponding components and solving the resulting linear system, the reader should be able to show that

so
Thus the transition matrix from B to is