DFT as an Orthogonal Basis Change
Hint 1: you can rewrite products of the exponential argument:

$$ f_{k,l}=\frac{1}{N}\left(e^{\frac{j2\pi(k-1)}{N}}\right)^{l-1}$$

and recognize a Vandermonde matrix. 

Hint 2: consider the dimension of the subspace spanned, and the linear independence.