Invariant subspaces under T and polynomials

Question

I am trying to solve the following problem from Hoffman and Kunze's linear algebra book.

Let $V$ be a vetor space over the field $F$, let $T$ be a linear operator on $V$, and let $f$ be a polynomial over the field $F$. If $W$ is the null space of $f(T)$, prove that $W$ is invariant under $T$.

I was not able to come up with any ideas, also, at this point, the book has not yet discussed minimal or characteristic polynomials, so I'd appreciate a hint for a solution that doesn't use these.

Answer 1

If $f(T) w = 0$ then $f(T) T w = T f(T) w = 0$.