Exercise 8, § 9

Let $E$ be a left affine space over a field $K$, $T$ its space of translations. a) In order that a mapping $f$ of $E$ into a left vector space $L$ over $K$ be affine, it is necessary and sufficient that one have

for all $\lambda \in K, t \in T, x, y$ in $E$. Let $x \mapsto [x]$ be the canonical injection of $E$ into the vector space $K_s^{(E)}$ of formal linear combinations of the elements of $E$, and let $N$ be the subspace of $K_s^{(E)}$ generated by the elements for $\lambda \in K, t \in T, x, y$ in $K$; finally, let $V$ be the quotient space of $K_s^{(E)}$ by $N$, and let $\varphi$ be the mapping of $E$ into $V$ which associates to every $x \in E$ the class of $[x]$ modulo $N$. Then $\varphi$ is an affine mapping and for every affine mapping $f$ of $E$ into a left vector space $L$ over $K$, there exists one and only one linear mapping $g$ of $V$ into $L$ such that $f = g \circ \varphi$. b) Let $\varphi_0 : T \to V$ be the linear mapping associated with $\varphi$, so that $\varphi(t + x) - \varphi(x) = \varphi_0(t)$. Now prove that $\varphi$ (and consequently also $\varphi_0$) is injective (consider the mapping $x \mapsto x - a$ of $E$ into $T$ for an $a \in E$); for every family $(\lambda_i)_{i \in I}$ of elements of $K$ with finite support, and every family $(x_i)_{i \in I}$ of elements of $E$, one has

Deduce that $\varphi_0(T)$ is a hyperplane passing through 0 in $V$, and $\varphi(E)$ a hyperplane parallel to $\varphi_0(T)$.