Exercise 10, § 2

Give an example of an A-linear mapping $u : E \to F$ such that ${}^t u$ is bijective but $u$ is neither injective nor surjective (cf. Exercise 6(b) and (c)).

Deduce an example where $y_0 \in F$ is orthogonal to the kernel of $t u$ but where the equation $u(x) = y_0$ has no solution.