If one is given a space of polynomials $P_n$ all of degree $n$ or less;
let $D$ be an operator as $D(p(x)) = \frac{dp}{dx}$ for all $p(x) \in P_n$
Identifying the polynomial $P(x) = a_0 + a_1x + ... + a_nx^n$ with the vector $[a_0, a_1, a_2, \dots, a_n]$ (which is the vector of $p(x)$ relative to the standard basis $\{1,x,x^2,...,x^n\}$) how could I compute the $(n+1)\times(n+1)$ matrix $D$?
Just transform all basis vectors $e_k=x^k$. Transform the first basis vector (p(x)=1) $$De_0=D(1)=0$$ First column is therefore empty.
$$De_1=D(x)=1=e_0$$ Second column is (1,0,0,0,0...). $$De_2=D(x^2)=2x=2e_1$$ Third column is (0,2,0,0,0...).
I think the pattern is obvious. The entire matrix is zero except one above the diagonal, where you get a sequence of positive integers (1,2,3,...).
In notation:
$$D_{ij}=\begin{cases}i & j=i+1\\0&\text{else} \end{cases}$$
Here I assumed the 1-based indexing (so $e_0=1=x^0$ will be at $i=1$). Sorry for the confusion but in the first part of the post I preferred the indices to match the powers.