Proposition 3

Let E, F be two right (resp. left) A-modules and $(e_i)_{i \in I}$, $(f_k)_{k \in K}$ finite bases of E and F respectively. For every linear mapping u of E into F, let $M(u)$ be the matrix of u with respect to the bases $(e_i)$ and $(f_k)$. Then the matrix of ${}^t u : F^* \to E^*$ with respect to the dual bases $(f_k^*)$ and $(e_i^*)$ is equal to ${}^t M(u)$.

E is canonically identified with its bidual $E^{**}$ and $(e_i)$ with the dual basis of $(e_i^*)$; then (supposing for example that E and F are right modules)

whence the proposition.