Let $f: V \to V$ be a linear map and $V = U \oplus U^\perp$ be an inner product space.
For its restriction map $f|_U$, is it invariant under $U$? And thus similarly for $f|_{U^\perp}$?
I guess one would try to see where the inner product $( ,)$ would map $f(u)$, but I don't seem to be able to get a good result.