$a)$ $T: P_3(R)\rightarrow P_3(R)$ given by $\ T(p(x))=xdp(x)/dx$
$b)$ $T: P_2(R)\rightarrow R^3$ given by $T(p(x))=(p(0), p(1), p(2))$
The precondition I know is $\dim (V)=\dim (W) \iff $the transformation has a isomorphism $\iff$ the transformation is both injective and surjective. (Correct?)
For $(a)$, how do we express the transformation in terms of matrices? Do we assume $p(x)$ to be $1+x+...+x^n$ so the matrix becomes $\begin{bmatrix}1&0&0&...\\0&1&0\\0&0&1\\.\\.\\.\end{bmatrix}$? And after differentiation it becomes $\begin{bmatrix}0&1&0&...\\0&0&1\\0&0&0\\.\\.\\.\end{bmatrix}$
For $(b)$, I have the same question from $(a)$.
If they are isomorphism, how to find the inverse?
A linear transformation between finite dimensional vector spaces of equal dimension is an isomorphism if and only if it is injective if and only if it is surjective. So it suffices to check only one of these.
HINTS: For (a) can you show if it is surjective or not? For (b) remember that a linear transformation is injective if and only if its kernel is trivial; with that what can you say about degree 2 polynomials such that $p(0)=p(1)=p(2)=0$?