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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.

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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?

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  • $\begingroup$ Do you mean the invariant transformation beyond r is determined by v? Is that why you complete the basis for V? Could you explain more in detail? I don't quite understand the idea of this problem. $\endgroup$ – clement Sep 29 '18 at 19:38

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