Page 187 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 171

Example 6.2.9
Consider two ordered bases of the vector space Pa(R):

2
B = {l,x,x } and B' = {1, 2x, Ax 2 - 2}.

In order to find the matrix S which describes the change of
basis B' —> B, we must write down the coordinate vectors of
the elements of B' with respect to the standard basis B: these
are

[l]s = I 0 , [2x] B = 2 J , [Ax 2 - 2] B = I 0

Therefore

The matrix which describes the change of basis B —> B' is

/ l 0 1/2'
S" 1 = 0 1/2 0
\ 0 0 1/4

For example, to express / = a + bx + ex 2 in terms of the basis
B', we compute

_1
[/]B'=S [/]fl

2
Thus / = (a + c/2)l + (b/2)2x + (c/4)(4x - 2), which is of
course easy to verify.