Following problem:
Let $\mathbb{K}$ be a Field and
$M = \begin{pmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ 1 & x_n & \ldots & x_n^{n-1} \end{pmatrix} \in Mat(n, \mathbb{K})$
Proove with induction that: $$\det(M) = \prod_{1 \le i < j \le n} (x_j - x_i)$$
Unfortunately I am totally lost here. I have confirmed $n = 1$ and for the rest I got the following:
$$\prod_{1 \le i < j \le n + 1} (x_j - x_i) = \prod_{1 \le i < j \le n} (x_j - x_i) \prod_{1 \le i < n + 1} (x_{n+1} - x_i) = \det(M_n) \prod_{1 \le i < n + 1} (x_{n+1} - x_i) $$
From there on I don't know how to continue.
Any ideas? :S Thanks!