Exercise 10, § 2

Give an example of an A-linear mapping $u : E \to F$ such that $^t u$ is bijective, but that $u$ is neither injective nor surjective (cf. II, p. 183, 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.