rank of a rectangular Vandermonde matrix

Question

Let the $m\times (n+1)$ rectangular Vandermonde matrix be $V$. More specifically, the matrix $V$ has the following form.

$V=\begin{pmatrix} 1 & a_1 & \cdots & a_1^{n} \\ 1 & a_2 & \cdots & a_2^{n}\\ \vdots& \vdots & \ddots &\vdots \\ 1 & a_m & \cdots & a_m^{n} \end{pmatrix}$ , where $m \geq n$.

I want a proof that $ \operatorname{rank}(V)=n$, if and only if the $a_i$ take exactly $n$ different values. Can you recommend me any paper or book that has a formal proof?

Answer 1

We will prove: If there are at least $k$ distinct values of the $a_i$ the rank is at least $k$.

Applying this for $k=n$ and $k=n+1$ gives the desired characterization of rank $n$.

Proof of the claim:

If there are less than $k$ distinct values there are less than $k$ distinct rows and the rank is clearly less than $k$.

If there are at least $k$ distinct values we can take the submatrix consisting of $k$ rows corresponding to distinct values and the first $k$ columns.

Its determinant is $$\prod_{1\le r < s \le k}(a_{i_r}-a_{i_s}) $$ which is nonzero so the rank is at least $k$.