Let $\mathrm{c}_{0}, \ldots, c_{n}$ be pairwise disctinct complexes. I proved that $\left(\left(X-c_{i}\right)^{n}\right)_{0 \leq i \leq n}$ is a basis of $\mathbb{C}_{n}[X]$ by induction.
1) Do you know another way to prove that those are linearly independent elements?
2) How would you prove it's a set of generators of $\mathbb{C}_{n}[X]$? ( without using 1, obviously)
3) I can't find the decomposition of 1 on this basis, I tried with the matrix method, but I couldn't conclude.