Exercise 22, § 7

Let $E$ and $F$ be two vector spaces over a field $K$ and $u$ a linear mapping of $E$ into $F$. If $V$ is a vector subspace of $E$ and $V'$ the orthogonal of $V$ in $E^*$, show that the dual of $u(V)$ is isomorphic to ${}^t u(F^*)/(V' \cap {}^t u(F^*))$. If $W'$ is a subspace of $F^*$ such that ${}^t u(W')$ is finite-dimensional and $W$ is the orthogonal of $W'$ in $F$, ${}^t u(W')$ is isomorphic to the dual of the space $u(E)/(W \cap u(E))$.