This problem comes from Section 2.4, Number 4, of Linear Algebra by Friedberg, Insel, and Spence.
The bases for and are, respectively, given as:
We are asked to compute the matrix form of the transformation,
Solution: I transformed each matrix, but this gave me 4 coefficient vectors, which means I end up with a 3x4 matrix. That means the input vector would have to be 4x1. Is that correct? How come the input isn't a 2x2 matrix?
EDIT again: the matrix I ended up with was:
EDIT: hit submit, rather than preview