I am trying to prove that a low-rank approximation of a matrix is not affected by left-multiplication by an orthogonal matrix. However, I am having problems with proving the following theorem:
Given $Q\in M_{m\times n}(\mathbb{R})$ matrix with orthogonal columns and $W\in M_{n\times r}(\mathbb{R})$ matrix of full rank, where $r\leq n\leq m$, show that: $$[QW]_k=Q[W]_k$$
In the given theorem, $[A]_k$ means the low-rank approximation of $A$ with rank $k$. Meaning, for $A=U\Sigma V^T$ and $u_i,v_i$ the columns of $U,V$, we get: $$[A]_k=\sum_{i=1}^k\sigma_iu_iv_i^T$$
How can I prove this theorem? Thanks for the help.