I need help solving the following proposition:
Let $f:V \to V$ be an endomorphism on a finite dimensional $K$-vector space $V$. Suppose the minimal polynomial of $f$, $p_f$, is a prime power, i.e. $p_f = p^r$ for some prime polyonomial $p\in K[x], r> 0$. Let $u\in V$ such that $p^{r-1}(f)(u) \neq 0$ and let $$U:= \langle u , f(u), f^2(u), \dots \rangle$$ be the $f$-cyclic subspace generated by $u$. Show that there exists a $f$-invariant subspace $U' \subset V$ such that $V = U \oplus U'$.
My attempt: Let $U'$ be a complement of $U$. Let $v\in U'$ and supppose $f(v)\in U.$ Then $$f(u) = \sum_{k=0}^{r-1}a_k f^k(u).$$ Applying $f^{r-1}$ on both sides yields $a_0 = 0$. Does this lead me anywhere? Is there something trivial I don't see? Any help appreciated!