Give a proof of Theorem 4.2 modeled on the proof of Proposition 3.2

QUESTION:

Give a proof of Theorem 4.2 modelled on the proof of Proposition 3.2.

Theorem 4.2:

Let V and W be finite-dimensional vector spaces, and let be a linear transformation. Let  and be ordered bases for . Let and be the change-of-basis matrices from  to and from  to, respectively. If and then, we have .

Proposition 3.2 (Change-of-Basis Formula, Take 1):

Let be a linear transformation with standard matrix. Let  be an ordered basis for and let be the matrix for with respect to . Let  be the matrix whose columns are given by the vectors. Then, we have

.