Page 258 - Matrix Analysis & Applied Linear Algebra

4.8 Change of Basis and Similarity 253

2
In this case, x 1 =1, x 2 = t, and x 3 = t , and y 1 =1, y 2 =1 + t, and
2
y 3 =1 + t + t , so the coordinates [x i ] B are computed as follows:
2
1 = 1(1) + 0(1 + t)+0(1 + t + t )= 1y 1 +0y 2 +0y 3 ,
2
t = − 1(1) + 1(1 + t)+0(1 + t + t )= −1y 1 +1y 2 +0y 3 ,
2
2
t = 0(1) − 1(1 + t)+1(1 + t + t )= 0y 1 − 1y 2 +1y 3 .
Therefore,

 
1 −1 0

P = [x 1 ] B [x 2 ] B [x 3 ] B =  0 1 −1  ,

0 0 1
2
and the coordinates of q = q(t)=3 + 2t +4t with respect to B are

     
1 −1 0 3 1
2
[q] B = P[q] B =  0 1 −1    =  −2  .
0 0 1 4 4
To independently check that these coordinates are correct, simply verify that

2
q(t) = 1(1) − 2(1 + t)+4(1 + t + t ).

It’s now rather easy to describe how the coordinate matrix of a linear oper-
ator changes as the underlying basis changes.

Changing Matrix Coordinates

Let A be a linear operator on V, and let B and B be two bases for
V. The coordinate matrices [A] B and [A] B are related as follows.

[A] B = P −1 [A] B P, where P =[I] BB (4.8.5)

is the change of basis matrix from B to B . Equivalently,

[A] B = Q −1 [A] B Q, where Q =[I] B B = P −1 (4.8.6)

is the change of basis matrix from B to B.