Relation between squared norms and sets of orthonormal vectors having the same span I was asked to show the following claim but I'm stuck. It seems I can't find the right reasoning path.
Given a matrix $M\in\mathbb{R}^{n\times d}$ and two sets of pairwise orthogonal unit vectors {$u_1, ..., u_k$} and {$v_1,..., v_k$} s.t. $span(\{u_1,...u_k\})=span(\{v_1,...v_k\})$. Show that $$\sum_{i=1}^{k}||Mu_i||^2=\sum_{i=1}^{k}||Mv_i||^2$$
Can anyone give me any hint?

Hints:
a) $v=Ru$ with $R$ orthogonal matrix
b) $||Mv||^2 = v^TM^TMv $
c) $Trace(R^T M^T M R) = Trace(M^T M)$
