Exercise 19, § 7

Let E be a vector space over a field K.

(a) Show that every mapping $f$ of E into E which is permutable with every automorphism of E is of the form $x \mapsto \alpha x$, where $\alpha \in K$ (show first that for all $x \in E$ there exists $\rho(x) \in K$ such that $f(x) = \rho(x)x$, using the fact that $f$ commutes with every automorphism of E leaving $x$ invariant).

(b) Let $g$ be a mapping of $E \times E$ into E such that, for every automorphism $u$ of E, $g(u(x), u(y)) = u(g(x, y))$ for all $x, y$ in E. Show that there exist two elements $\alpha, \beta$ of K such that $g(x, y) = \alpha x + \beta y$ on the set of ordered pairs $(x, y)$ of linearly independent elements of E; moreover there exists a mapping $\phi$ of $K \times K$ into K such that $g(\lambda x, \mu x) = \phi(\lambda, \mu)x$ for all $x \in E$ (same method). If further $g(u(x), u(y)) = u(g(x, y))$ for every endomorphism $u$ of E, then $g(x, y) = \alpha x + \beta y$ for all $x, y$ in E. Generalize to mappings of $E^n$ into E.