# A polynomial basis of Cn[X]

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.

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