Proposition 6

Let $u$ be an affine mapping of $E$ into $E'$. There exists one and only one linear mapping $v$ of $T$ into $T'$ such that

for all $x \in E, t \in T$.

Let $a$ be any point of $E$. The mapping

is a linear mapping $v_a$ of $T$ into $T'$, for we may write

and it follows from (3) that $v_a(\lambda t) = \lambda v_a(t)$ and $v_a(s + t) = v_a(s) + v_a(t)$. Moreover, if $b$ is another point of $E$, then $v_a = v_b$; for the relation

implies

that is $u(a + t) - u(a) = u(b + t) - u(b)$. Whence the existence of $v$; the uniqueness is immediate.

$v$ is called the linear mapping of $T$ into $T'$ associated with $u$. Conversely, for every linear mapping $v$ of $T$ into $T'$ and every ordered pair of points $a \in E, a' \in E'$, it is immediately verified that

is an affine mapping of E into E' whose associated linear mapping is v. To say that u is an affine mapping of E into E' therefore also means that, if an arbitrary point a in E and the point u(a) in E' are taken as origins, u is a linear mapping for the two vector spaces thus obtained.

Let E'' be a third affine space, T'' its translation space, u' an affine mapping of E' into E'' and v' the linear mapping of T' into T'' associated with u'. Clearly u' o u is an affine mapping of E into E''; moreover, for a \in E and t \in T,

and hence v' o v is the linear mapping of T into T'' associated with u' o u. For an affine mapping u to be bijective, it is necessary and sufficient that the associated linear mapping v be so, and u^{-1} is then an affine mapping whose associated linear mapping is v^{-1}.

In particular, the affine bijections of E onto itself form a group G, called the affine group of E. The mapping which associates with u \in G the linear mapping v associated with u is, by the above, a homomorphism of G onto the linear group \mathbf{GL}(T). If u is a translation, v is the identity and conversely. Hence, the kernel of the above homomorphism is the translation group T of E which is therefore a normal subgroup of G.

If u \in G, the automorphism t \mapsto utu^{-1} of T (where t is identified with the translation x \mapsto x + t) is the linear mapping v associated with u. For x \in E and t \in T, by definition

and hence utu^{-1} = v(t).

Let a \in E and G_a be the subgroup of G consisting of the u \in G such that u(a) = a. If E is identified with T by taking a as origin, G_a is identified with \mathbf{GL}(T). Every u \in G can be expressed uniquely in the form u = t_1u_1 (resp. in the form u = u_2t_2), where u_1, u_2 are in G_a and t_1, t_2 in T: for, writing t_1 = u(a) - a, u^{-1}t_1 \in G_a, whence the existence of u_1 and t_1; the existence of u_2 and t_2 is obtained analogously. The uniqueness follows from the fact that G_a \cap T reduces to the identity element of G. Moreover

whence u_2 = u_1, t_2 = u_1^{-1}t_1u_1. Finally, the linear mappings associated with u and u_1 are the same and hence, if as above G_a is identified with \mathbf{GL}(T), u_1 is the linear mapping from T to itself associated with u. It is thus seen that G is the semi-direct product of G_a by T (I, § 6, no. 1).

Let E, E' be two affine spaces over K. The direct (resp. inverse) image of a linear variety of E (resp. E') under an affine mapping u of E into E' is a linear variety of E' (resp. E); the rank of u is by definition the dimension of u(E); it is equal to the rank of the linear mapping associated with u. If V, V' are linear varieties of the same finite dimension m in E, E' respectively, there exists an affine mapping $u$ of $E$ into $E'$ such that $u(V) = V'$: taking as origins in $E$ and $E'$ points of $V$ and $V'$ respectively, then taking in $E$ (resp. $E'$) a basis whose first $m$ vectors form a basis of $V$ (resp. $V'$), the proposition follows immediately from § 1, no. 11, Corollary 3 to Proposition 17.

As the field $K$ has canonically a left vector space structure (of dimension 1) over $K$, it can be considered as an affine space of dimension 1. An affine mapping of an affine space $D$ (over $K$) into the affine space $K$ is also called an affine linear function (or an affine function). If a point $a$ is taken as origin in $E$, every affine function on $E$ can then be written uniquely as $x \mapsto \alpha + v(x)$, where $\alpha \in K$ and $v$ is a linear form on the vector space $E$ thus obtained; the affine functions on $E$ therefore form a right vector space over $K$ of dimension $1 + \dim E$. If $u$ is a non-constant affine function on $E$ and $\lambda \in K$, the set of $x \in E$ satisfying the equation $u(x) = \lambda$ is a hyperplane; conversely, for every hyperplane $H$ in $E$, there exists an affine function $u_0$ on $E$ such that $H = u_0^{-1}(0)$ and every affine function $u$ such that $H = u^{-1}(0)$ is of the form $u_0 \mu$, where $\mu \in K$ (§ 7, no. 5, Proposition 11). If $u$ is an affine function on $E$, the hyperplanes with equations $u(x) = \alpha$ and $u(x) = \beta$ are parallel.