Let \(k\) be a field and let \(V\) and \(W\) be finite-dimensional \(k\)-vector spaces. The \(k\)-linear map \[ \begin{aligned} \Phi\colon \dualof{V}\tensor_k W &\to \Hom_k(V,W),\\ \tilde v\tensor w &\mapsto \big(v\mapsto \tilde v(v)\,w\big), \end{aligned} \] is an isomorphism.
The map \(\dualof{V}\times W\to\Hom_k(V,W)\), \((\tilde v,w)\mapsto \tilde v(\wait)\,w\), is \(k\)-bilinear, so \(\Phi\) is well defined by the universal property of the tensor product. Let \(e_1,\ldots,e_n\) be a basis of \(V\) with dual basis \(e_1^*,\ldots,e_n^*\) of \(\dualof{V}\), and let \(f_1,\ldots,f_m\) be a basis of \(W\). The elements \(e_i^*\tensor f_j\) form a basis of \(\dualof{V}\tensor_k W\). The map \(\Phi(e_i^*\tensor f_j)\) sends \(e_i\) to \(f_j\) and \(e_l\) to \(0\) for \(l\neq i\); in these bases its matrix is the matrix unit \(E_{ji}\). The matrix units \(E_{ji}\) form a basis of \(\Hom_k(V,W)\), so \(\Phi\) sends a basis to a basis and is an isomorphism.