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Let $V$ be a finite dimensional vector space over a field $F$ and let $T:V\to V$ be a linear transformation. Let $W\subseteq V$ be a subspace such that $T(W)\subseteq W$. Suppose $T$ is diagonalizable. Is $T$ restricted to $W$ also diagonalizable?

I start from the fact that a LT is diagonalizable if it has all distinct eigenvalues. Then the contrapositive states that a LT is not diagonalizable implies it has at least two equal eigenvalues. Now I want to use that fact to show some connection with the dimension of $W$ and $V$, but I've not been successful. Am I going a wrong way here?

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