TITLE: Proof $\langle v,w\rangle=0\implies \langle Av,Aw\rangle=0$, given $A\in F^{m\times n}, \operatorname{rank}(A)=n$.
QUESTION [3 upvotes]: Given two vectors $v,w$ orthogonal, and a matrix $A$ which has orthogonal columns, how to prove that $\langle Av,Aw \rangle=0$, i.e. their image is orthogonal?

REPLY [1 votes]: You have $$\langle Av,Aw\rangle=\langle v,A^TAw\rangle=\langle v,Id.w\rangle=\langle v,w\rangle=0$$by definition of the transpose of $A$ and because $A$ is supposed to be orthogonal.
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