Exercise 1, § 2

Let K be a commutative ring with unit element, let E be an A-module, and let $E^*$ be its dual. Denote by $\varphi$ the canonical homomorphism from $E \otimes E^*$ to $\mathrm{End}(E)$.

a) Any element distinct from 1 in $\mathrm{End}(E)$ of the form

with $x \in E$ and $y^* \in E^*$, is called a pseudo-reflection in E. Such an element s is called a reflection if $x$ and $y^*$ can be chosen so that $\langle x, y^* \rangle = 2$; show that we then have $s^2 = 1$ and $s(x) = -x$.

b) Let $x \in E, \ y^* \in E^*$ be such that $\langle x, y^* \rangle = 1$. Show that E is the direct sum of the submodule $Kx$ generated by $x$ and the orthogonal complement H of $y^*$. Show that $Kx$ is free with basis $x$, and that $s = s_{2x, y^*}$ is equal to 1 on H and to $-1$ on $Kx$.