Direct sum of $T - invariant$ subspaces

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 ?

If $W$ is a proper subspace then take $f(X)=X$ then your question is not true, as then $V=W$ is coming from the equation we are needed to prove, $V$ is not the direct sum of those $T$ invariant subspaces, $W$ is the direct sum!!
• I edited the question, $W$ is $T - invariant$ as well – user401516 May 10 '17 at 20:26