Let $a=(a_0,$dots,$a_{n-1})$ be a real vector and let $b=(b_0,$dots,$b_{n-1})$ be its

discrete Fourier transform. Prove that

$n{\displaystyle \underset{i=0}{\overset{n-1}{}}}{a}_i^2={\displaystyle \underset{i=0}{\overset{n-1}{}}}|b_i{|}^2.$