Exercise 8, § 9

Let E be a left affine space over a field K and T its translation space.

(a) For a mapping f of E into a left vector space L over K to be affine, it is necessary and sufficient that

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 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 $E$; finally, let $V$ be the quotient space of $K_s^{(E)}$ by $N$ and $\phi$ the mapping of $E$ into $V$ which associates with every $x \in E$ the class of $[x]$ modulo $N$. Then $\phi$ 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 $f = g \circ \phi$.

(b) Let $\phi_0 : T \to V$ be the linear mapping associated with $\phi$, so that $\phi(t + x) - \phi(x) = \phi_0(t)$. Show that $\phi$ (and therefore also $\phi_0$) is injective (consider the mapping $x \mapsto x - a$ of $E$ into $T$ for some $a \in E$); 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 $E$,

Deduce that $\phi_0(T)$ is a hyperplane passing through 0 in $V$ and $\phi(E)$ a hyperplane parallel to $\phi_0(T)$.