Let $T$ be a linear operator on the vector space $V$ over the field $F$. If $f$ is a polynomial over $F$ and $a$ is in $V$, let $fa = f(T)a$. If $V_1, . . . , V_k$ are $T$-invariant sub-spaces and $V = V_1\oplus V_2\oplus....\oplus V_k$, show that $f V =f V_1\oplus f V_2\oplus...\oplus f V_k$.
Let $a = v_1 + v_2 +...+ v_k$ since $V_i's$ are T-invariant we have $Tv_i \in V_i$ and hence $fv_i = f(T)V_i \subset fV_i$. Can we conclujde from here...Please tell if my lolgic is wrong.