Page 257 - Matrix Analysis & Applied Linear Algebra

252 Chapter 4 Vector Spaces

Changing Vector Coordinates
Let B = {x 1 , x 2 ,..., x n } and B = {y 1 , y 2 ,..., y n } be bases for V,

and let T and P be the associated change of basis operator and change
of basis matrix, respectively—i.e., T(y i )= x i , for each i, and

P =[T] B =[T] B =[I] BB = [x 1 ] B [x 2 ] B ··· [x n ] B . (4.8.3)

• [v] B = P[v] B for all v ∈V. (4.8.4)
• P is nonsingular.
• No other matrix can be used in place of P in (4.8.4).

Proof. Use (4.7.6) to write [v] B =[I(v)] B =[I] BB [v] B = P[v] B , which is
(4.8.4). P is nonsingular because T is invertible (in fact, T −1 (x i )= y i ), and
because (4.7.10) insures [T −1 ] B =[T] −1 = P −1 . P is unique because if W is
B
another matrix satisfying (4.8.4) for all v ∈V, then (P − W)[v] B = 0 for all
v. Taking v = x i yields (P − W)e i = 0 for each i, so P − W = 0.

If we think of B as the old basis and B as the new basis, then the change

of basis operator T acts as
T(new basis) = old basis,

while the change of basis matrix P acts as

new coordinates = P(old coordinates).
For this reason, T should be referred to as the change of basis operator from

B to B, while P is called the change of basis matrix from B to B .
Example 4.8.1
Problem: For the space P 2 of polynomials of degree 2 or less, determine the
change of basis matrix P from B to B , where

2 2

B = {1,t,t } and B = {1, 1+ t, 1+ t + t },
2
and then find the coordinates of q(t) = 3+2t +4t relative to B .

Solution: According to (4.8.3), the change of basis matrix from B to B is

P = [x 1 ] B [x 2 ] B [x 3 ] B .