Exercise 25, § 1

Let E, F be two A-modules and $u: E \to F$ a linear mapping. Show that the mapping $(x, y) \mapsto (x, y - u(x))$ of the product module $E \times F$ to itself is an automorphism of $E \times F$. Deduce that if there exists a linear mapping $v: F \to E$ and an $a \in E$ such that $v(u(a)) = a$, there exists an automorphism $w$ of $E \times F$ such that $w(a, 0) = (0, u(a))$.