I have the following problem:
Let $(a_{1},\dots,a_{n})\in\mathbb{R}^{n}$ a vector and let the matrix (Vandermonde) \begin{equation*} V = \begin{pmatrix} 1&a_{1}&a_{1}^{2}&\dots&a_{1}^{n-2}&a_{1}^{n-1}\\ 1&a_{2}&a_{2}^{2}&\dots&a_{2}^{n-2}&a_{2}^{n-1}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 1&a_{n}&a_{n}^{2}&\dots&a_{n}^{n-2}&a_{n}^{n-1} > \end{pmatrix} \end{equation*} Prove that \begin{equation*} \text{det}(V) = \prod_{1\leq i < j \leq n}(a_{j}-a_{i}). \end{equation*}
What I have tried is, by induction: \begin{equation*} V_{2} = \begin{pmatrix} 1&a_{1}\\ 1&a_{2} \end{pmatrix} \Rightarrow \text{det}(V_{2}) = a_{2} - a_{1}, \end{equation*} Suppose it is true for $ n $. Let's try to test it for $ n + 1 $, let \begin{equation*} V_{n+1} = \begin{pmatrix} 1&a_{1}&a_{1}^{2}&\dots&a_{1}^{n-2}&a_{1}^{n-1}&a_{1}^{n}\\ 1&a_{2}&a_{2}^{2}&\dots&a_{2}^{n-2}&a_{2}^{n-1}&a_{2}^{n}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 1&a_{n}&a_{n}^{2}&\dots&a_{n}^{n-2}&a_{n}^{n-1}&a_{n}^{n}\\ 1&a_{n+1}&a_{n+1}^{2}&\dots&a_{n+1}^{n-2}&a_{n+1}^{n-1}&a_{n+1}^{n} \end{pmatrix}, \end{equation*} an indication is to use appropriate determinant properties and row operations.
