Let be a -dimensional vector space over a fixed field . A set is called an affine space of dimension iff it carries a free and transitive action of the additive group of the vector space .
Thus an affine space is formally a triple where is the action. We also write .
Let . Then by transitivity of the action, there is an element such that . By freeness such an element is unique so we denote that unique element by . Thus . Other immediate properties are , and what justifies skipping some brackets. A proof of the last equality goes as follows:
c + (b - a) = (b + (c - b)) + (b - a) = b + ((c-b) + (b-a)) = b + (c - a).
For each point we define a map by . This map is bijective and therefore there is a unique vector space structure on which makes an isomorphism of vector spaces. That vector space structure on depends on ; thus we will denote it by . For each pair we can therefore define vector space isomorphisms and .
Let and be two affine spaces. A map of sets is called an affine map if a linear map such that
A(a + v) = A(a) + L v, \,\,\forall v \in V. \,\,\,\,(1)
In other words, . That property is satisfied iff it is satisfied for a single . On the other hand each element can be represented as so that if we are given two points and a linear map then affine map such that the equation (1) holds and .
See also affine space.