Let $T:V \to V$ be a linear transformation, $W \subset V$ a $T - invariant$ subspace of $V$, and $f \in F[X] \;$ for some field $F$.
Given that $ \; W = W_1 \oplus \cdots \oplus W_k \; $ such that $W_i \;$ is $T - invarient$, I need to prove that:
$$f(T)(V)=f(T)(W_1) \oplus \cdots \oplus f(T)(W_k)$$
I managed to prove that $W_i $ is $f(T) -invariant$. I alsot proved that $f(T)(W_i)$ are disjoint subspaces. I'm trying to prove that $V = W_1 + \cdots + W_k$ with dimensions, but without success. Any ideas ?