There are some questions discussing the diagonalizability of a restriction of a diagonalizable endomorphism to an invariant subspace, however, I have a question regarding a certain approach, which wasn't covered by previous questions. (I'm currently brushing up on linear algebra, but there's one exercise I can't quite get my head around.)
Let $f:V\to V$ be a diagonalizable endomorphism of the finite-dimensional vector space $V$ over the field $K$. Let $W\subseteq V$ be a subspace, such that $f(W)\subseteq W$. Let $\bar f:V/W\to V/W$ be the induced map. Prove that $f|_W$ is diagonalizable by first proving that $\bar f$ is diagonalizable with eigenspaces $E(\bar f,a_i)=E(f,a_i)/E(f,a_i)\cap W$ and then using the definition $$f\text{ is diagonalizable}\ \Leftrightarrow\ \dim V=\sum_iE(f,a_i).$$
I know how to get from "$f$ and $\bar f$ are diagonalizable" to "$f|_W$ is diagonalizable", but (I need some practice with quotient spaces) I can't find a proof for $E(\bar f,a_i)=E(f,a_i)/E(f,a_i)\cap W$.