Zoran Skoda
affine space

Let V be a n-dimensional vector space over a fixed field k. A set is called an affine space of dimension n iff it carries a free and transitive action of the additive group of the vector space V.

Thus an affine space is formally a triple (,V,μ) where μ is the action. We also write a+v=defμ(v,a).

Let a,b. Then by transitivity of the action, there is an element vV such that b=a+v. By freeness such an element is unique so we denote that unique element by ba. Thus a+(ba)=b. Other immediate properties are aa=0, and c+(ba)=b+(ca) what justifies skipping some brackets. A proof of the last equality goes as follows:

c+(ba)=(b+(cb))+(ba)=b+((cb)+(ba))=b+(ca).c + (b - a) = (b + (c - b)) + (b - a) = b + ((c-b) + (b-a)) = b + (c - a).

For each point a we define a map ϕ a:V by ϕ a(b)=ba. This map is bijective and therefore there is a unique vector space structure on which makes ϕ a an isomorphism of vector spaces. That vector space structure on depends on a; thus we will denote it by V a. For each pair (a,b)× we can therefore define vector space isomorphisms ϕ ab=ϕ a 1ϕ b:V bV a and ψ ab=ϕ aϕ b 1:VV.

Let (,V,μ) and (,V,μ) be two affine spaces. A map of sets A: is called an affine map if a linear map L:VV such that

A(a+v)=A(a)+Lv,vV.(1)A(a + v) = A(a) + L v, \,\,\forall v \in V. \,\,\,\,(1)

In other words, (Aμ)(v,a)=μ(Lv,A(a)). That property is satisfied iff it is satisfied for a single a=p. On the other hand each element b can be represented as p+(bp) so that if we are given two points p,q and a linear map L:VV then ! affine map A: such that the equation (1) holds and A(p)=q.

See also affine space.

Revised on August 25, 2009 19:55:47 by Zoran Škoda (