Existence of matrix $C$ such that $f(A)=tr(AC)$

Question

Let $f: M_{n\times n}(\mathbb R)\to \mathbb R$ a functional. Prove that there exists a unique square matrix $C$ such that $f(A)=tr(AC)$ for all $A\in M_{n\times n}(\mathbb R)$

I´ve been trying to solve this problem but unfortunately I don´t know where to start. I would really appreciate if you can give me some hints or suggestions

Answer 1

Hint: Note that $\operatorname{trace}(AB^T)$ is just the "dot product" of $A$ and $B$.