For $x_0,...,x_n \in \mathbb{R}$ is
$V(x_0,...,x_n)=$\begin{bmatrix} 1 & x_{0} & x_{0}^{2} & ...........x_{0}^{n} \\ 1 & x_{1} & x_{1}^{2} & ...........x_{1}^{n} \\ . & x_{2} & x_{2}^2 & ...........x_{2}^{n} \\ . & . & . &...........x_{3}^{n} \\ . & . & . & ...........x_{4}^{n} \\ . & . & . & \\ . & . & .\\ . & . & .\\ 1 & x_n & x^{2}_n & ...........x_{n}^{n} \\ \end{bmatrix}
How can i show now that det $V(x_0,...,x_n) = \prod_{0 \leq i <j\leq n} (x_j - x_i)$? Would be enough for $n=0,1,2$ but a general approach would be nice. I know the proof of this but only starting at $x_1$. Could I do just an indexshift?
How can I with 1. now proof that wiht $n+1$ datapoints $(x_i,f_i)$ there is just one polynomial $p(x)$ with deg$\leq n$ (polynomial interpolation) And can that degree be any number between 0 and n? Are there examples?