I really need some help with this question:
Let $V = M_{n \times m}(\mathbb{K})$ be a vector space of matrices, where $\mathbb{K}$ is some field and $f \in V^*$ a linear functional. Prove that there is a matrix $P \in V$ with $f(A) = \mbox{trace}(P^T A)$ for all $A \in V$.
So I have a functional $f$ and have to construct a Matrix $P$. I think the basis of $V$,
$$B = \left\{ E_{ij} \mid 1 \leq i \leq n, 1 \leq j \leq m \right\}$$
will be useful but I couldn't figure out what to do next.