# F-Invariant matrix has a block triangular matrix

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.

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?