I've noticed that in most sources the definition of the DFT is given by $$F_k = DFT\{f_n\}=\frac{1}{N}\sum\limits_{n=0}^{N-1}f_nW_N^{-kn}$$ where $W_N$ is the $N$th root of unity and $0\le k \le N-1$.
While in other sources the definition is $$F_k = DFT\{f_n\} = \frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1}f_nW_N^{kn}$$
Are these definitions really equal? How is this justified?