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174 Chapter Six: Linear Transformationns
matrix A with respect to B and C, and by a matrix A' with
respect to B' and C, then
A' = YAX-\
The most important case is that of a linear operator
T : V —• V, when the ordered basis B is used for both domain
and codomain.
Theorem 6.2.6
Let B and B' be two ordered bases of a finite-dimensional vec-
tor space V and let T be a linear operator on V. If T is repre-
sented by matrices A and A' with respect to B and B' respec-
tively, then
A' = SAS' 1
where S is the matrix representing the change of basis B —> B'.
Example 6.2.11
Let T be the linear transformation on -Ps(R) defined by
/
T(/) = '. Consider the ordered bases of Pa(R)
2
B = {l,x,x } and B' = {l,2x,4x 2 - 2}.
We saw in Example 6.2.9 that the change of basis B —> B'
is represented by the matrix
/ l 0 1/2'
17 = 0 1/2 0
\ 0 0 1/4
Now T is represented with respect to B by the matrix
A
