How do I prove that for every monic polynomial $P$ there exists a linear operator $f$ such that $P$ is the minimal polynomial of $f$?

Question

Let $\mathbb{F}$ be a field and $P(T)=T^{k}+a_{k-1}T^{k-1}+\dots+a_{1}T+a_{0}\in\mathbb{F}[T]$ a monic polynomial.

Prove: There exist a vector space $V$ over $\mathbb{F}$, a linear operator $f:V\to V$ and a vector $v\in V $ such that the minimal polynomial is equal to $P$

$(min_{v}^{f}=P)$

Hint: use the Companion matrix of P

I'd appreciate some help, I've been stuck on this problem for a good while. Moreover, I can't quite figure out how to use the companion matrix in my proof.