The trace pairing identifies $\Hom(V,W)$ with $\dualof{\Hom(W,V)}$

Let \(k\) be a field and \(V, W\) finite-dimensional \(k\)-vector spaces. The map \[ \begin{aligned} \Hom_k(V, W) &\to \dualof{\Hom_k(W, V)} \\ T &\mapsto \big(S \mapsto \Tr(T \circ S)\big) \end{aligned} \] is an isomorphism of \(k\)-vector spaces, where \(T \circ S\colon W \to W\).