Let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct real numbers. For each $i$ let $\mathbf{v}_{i}$ be the vector $\left(1, a_{i}, a_{i}^{2}, \ldots, a_{i}^{n-1}\right)$. Let $A$ be the $n \times n$ matrix with rows $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}$ and let $\mathbf{c}$ be a column vector of size $n$. Prove that $A \mathbf{c}=\mathbf{0}$ if and only if $\mathbf{c}=\mathbf{0}$. Deduce that the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} \operatorname{span} \mathbb{R}^{n}$.

[You may use general facts about matrices if you state them clearly.]