F-Invariant matrix has a block triangular matrix

Question

Suppose $F : V \to V$ is linear. A subspace $W$ of $V$ is said to be invariant under $F$ if $F(W) \subseteq W$. Suppose $W$ is invariant under $F$ and $\mbox{dim}(W)=r$. Show that $F$ has a block triangular matrix representation $$M = \begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where A is an $r \times r$ submatrix.

Answer 1

Builds an orthonormal basis $\{w_i\}$ for $W$, and complete it to an orthonormal basis $\{w_1, \ldots, w_r, v_1, \ldots\}$ for all of $V$.

What do you know about the action of $M$ on a generic vector in $W$ expressed in this basis? What can you conclude about $M$'s representation in this basis?