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How to find matrix form of an operator on a vector space V which is direct sum of its two subspaces?
So Let's say I have a vector space $V$ over a field $F$ which is n dimensional, and additionally that I have $V=B\oplus C$ with $B$ and $C$ subspaces. If I have a linear transformation $T:V\rightarrow V$ such that restricted to either of these subspaces, $T$ is invariant, then obviously we have a matrix in $F^{n\times n}$ corresponding to $T$. I've seen the claim that if we form the matrix $Q$ whose left-most columns are a basis for $B$ and the rest a basis for $C$ that $Q^{-1}TQ$ is block diagonal. I've never quite understood why this was true, nor have I been able to prove it...Certainly this is important for proving the existence of Jordan Normal Forms etc, so I'd like to be able to justify the result.