Prove that Fourier matrix is invertible

Question

I need to show that the Fourier matrix is invertible using the fact that the determinant of the Vandermonde matrix is given by

$$ det(V_n) = \prod_{s,t=0 \\ s<t}^n (z_t - z_s). $$

It is not clear how I should do that but is it possible to prove it using the fact that it is a unitary matrix and so it is invertible?

Answer 1

Hint

The matrix $V$ is invertible if and only if $\det(V) \ne 0$. Take a look at the polynomial $$ \det(V_n) = \prod_{s,t=0 \\ s<t}^n (z_t - z_s). $$ and try to guess why $z_t - z_s$ is never zero.