Let $V=\mathbb{R}^{n}$ and $T\,:V\to V$ be defined by $Tv=Av$ where $A\in M_{n}(\mathbb{R})$ is an orthogonal matrix.
My lecture wrote that if $W\subset V$ is a subspace of $V$ then if $W$ is $A$ invariant then $W^{\perp}$ is also $A$ invariant.
What I do know is that if $W$ is $A$ invariant then $W^{\perp}$ is also $A^*=A^{t}$ invariant, but I could not deduce from this that it is also $A$ invariant.
Is this 'fact' true ? I couldn't prove it (I tried writing a proof similar to the case I know, using inner products and failed), help is appreciated!