Here is the problem:
How do I do this? Help please!!
For example, a good basis for P2 (the space of polynomials of degree at most 2) is $\displaystyle e_1= 1$, $\displaystyle e_2= x$, $\displaystyle e_3= x^2$. Apply the transformation to each of those and write the result as a linear combination of them. The numbers multiplying each vector in the linear combination will be one column of the matrix.
(When I first looked at this, I made a very foolish error. I assumed that, like the derivative operator itself, this would have non-trivial kernel and so would be non-invertible and have determinant 0. Fortunately, however, I chose to check that and discovered it was not true!)