Consider a complex vector subspace $W \subset V$ such that $\mathrm{dim}W = n$. Suppose I have a set $S$ of $n$ linearly independent complex matrices $S = \{ M_1, \cdots, M_n\}$ that act on $V$.
Let $w$ be some non-zero vector in $W$ and consider $$ w_1' = M_1w, \quad w_2' = M_2 w, \quad \cdots \quad w_n'= M_n w. $$ Is it always possible to choose a non-zero vector $w$ so that $w_1', \cdots, w_n'$ is a mutually orthogonal basis for $W$ with respect to the standard complex inner product? If so, is it possible to construct $w$?
(If linear independence is not strong enough, can we find such a $w$ if $S$ is instead mutually orthogonal with respect to the trace inner product $\langle A, B \rangle = \mathrm{Tr}(A^\dagger B)$?)