I know pretty well how to find the transformation matrix of a linear map (with respect to a basis). However, I am wondering whether it is also possible to do it the other way around?
This question arises because in one of my exercises in linear-algebra I first had to find the transformation matrix with respect to the canonical basis to the linear map F: \mathbb{R}_2[t] \to \mathbb{R}_2[t],\; s(t) \mapsto s(t) + s'(t) + ts''(t), then I have to find the inverse map $F^{-1}$. The sample solution only provides the inverse of the transformation matrix, however, I want to know whether I can give an explicit function (using the inverse of the transformation matrix).
Thanks for any help in advance!