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.

  • 1
    $\begingroup$ The answer to question 2 is immediate after 1.... $\endgroup$ – Euler Pythagoras Aug 17 at 20:04
  • $\begingroup$ I want a method for 2 without using 1, obviously. $\endgroup$ – shocop Aug 17 at 20:06
  • $\begingroup$ Do you know anything about linear algrebra? $\endgroup$ – Euler Pythagoras Aug 17 at 20:07
  • $\begingroup$ More then you could imagine :) $\endgroup$ – shocop Aug 17 at 20:09
  • $\begingroup$ 3) is indeed solved easily using these matrices. $\endgroup$ – metamorphy Aug 17 at 20:10

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