Vandermonde matrix rank

It suffices to show that the first $N$ rows are linearly independent. In this regard, you may see ProofWiki for two different proofs.

You can easely prove by induction that

$$\det \begin{bmatrix}1&1&\cdots&1 \\ z_1&z_2&\cdots&z_N\\ \vdots&\vdots&\ddots&\vdots\\ z_1^{N-1}&z_2^{N-1}&\cdots&z_N^{N-1} \end{bmatrix} = \prod_{1 \leq i< j \leq N}(Z_j-Z_i)$$

Thus, if the $z_i$ are pairwise distinct, this determinant is non-zero, which shows that the first $N$ rows of your matrix are linearly independent.

You need to be careful. Although the determinant is never zero, the condition number of Vandermonde matrices tends to be very large.