Page 189 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 173

Change of basis and linear transformations

We are now in a position to calculate the effect of change
of bases on the matrix representing a linear transformation.
Let B and C be ordered bases of finite-dimensional vector
spaces V and W over the same field, and let T : V —> W be
a linear transformation. Then T is represented by a matrix A
with respect to these bases.
Suppose now that we select new bases B' and C for V
and W respectively. Then T will be represented with respect
to these bases by another matrix, say A'. The question before
us is: what is the relation between A and A'l
Let X and Y be the invertible matrices that represent
the changes of bases B —> B' and C —> C respectively. Then,
for any vectors v of V and w of W, we have

[v] B/ = X[V]B and [w] C/ = y[w] c .
Now by 6.2.3

[T(v)] c = A[w) B and [T(v)] c, = A'[v] s ,.

On combining these equations, we obtain

[T(v)] c , = Y[T(v)] c = 7A[v] B = YAX-^B'-

But this means that the matrix YAX -1 describes the linear
transformation T with respect to the bases B' and C of V and
l
W respectively. Hence A' = YAX~ .
We summarise these conclusions in

Theorem 6.2.5
Let V and W be non-zero finite-dimensional vector spaces over
the same field. Let B and B' be ordered bases of V, and C
and C ordered bases of W. Suppose that matrices X and Y
describe the respective changes of bases B —> B' and C —> C'.
If the linear transformation T : V —> W is represented by a