Exercise 5, § 1

Let V and W be vector spaces of dimensions $n + 1$ and $n$ over a field K. Show that $\mathfrak{gl}(W)$ is isomorphic to a subalgebra of $\mathfrak{sl}(V)$. (W can be assumed to be a hyperplane of V. Let $e \in V, e \notin W$. For $x \in \mathfrak{gl}(W)$, let $y(x)$ be the element of $\mathfrak{gl}(V)$ which extends $x$ and is such that $y(e) = -\operatorname{Tr}(x) \cdot e$. Show that the mapping $x \mapsto y(x)$ is an isomorphism of $\mathfrak{gl}(W)$ onto a subalgebra of $\mathfrak{sl}(V)$.