Affine space

over a field A set (the elements of which are called the points of the affine space) to which corresponds a vector space over (which is called the space associated to ) and a mapping of the set into the space (the image of an element is denoted by and is called the vector with beginning in and end in ), which has the following properties:

a) for any fixed point the mapping , , is a bijection of on ;

b) for any points the relationship where denotes the zero vector, is valid. The dimension of is taken for the dimension of the affine space . A point and a vector define another point, which is denoted by , i.e. the additive group of vectors of the space acts freely and transitively on the affine space corresponding to .

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Examples.

1) The set of the vectors of the space is the affine space ; the space associated to it coincides with . In particular, the field of scalars is an affine space of dimension 1. If , then is called the -dimensional affine space over the field , and its points and determine the vector .

2) The complement of any hyperplane in a projective space over the field is an affine space.

3) The set of solutions of a system of linear (algebraic or differential) equations is an affine space the associated space of which is the space of solutions of the corresponding homogeneous set of equations.

A subset of an affine space is called an affine subspace (or a linear manifold) in if the set of vectors , , forms a subspace of . Each affine subspace has the form , where is some subspace in , while is an arbitrary element of .

A mapping between affine spaces and is called affine if there exists a linear mapping of the associated vector spaces such that for all , . A bijective affine mapping is called an affine isomorphism. All affine spaces of the same dimension are mutually isomorphic.

The affine isomorphisms of an affine space into itself form a group, called the affine group of the affine space and denoted by . The affine group of the affine space is denoted by . Each element is given by a formula where  being an invertible matrix. The affine group contains an invariant subgroup, called the subgroup of (parallel) translations, consisting of the mappings for which is the identity. This group is isomorphic to the additive group of the vector space . The mapping defines a surjective homomorphism of into the general linear group GL, with the subgroup of parallel translations as kernel. If is a Euclidean space, the pre-image of the orthogonal group is called the subgroup of Euclidean motions. The pre-image of the special linear group SGL is called the equi-affine subgroup (cf. Affine unimodular group). The subgroup consisting of the mappings such that for a given and arbitrary is called the centro-affine subgroup; it is isomorphic to the general linear group GL of the space .

In algebraic geometry an affine algebraic set is sometimes called an affine space. A finite-dimensional affine space can be provided with the structure of an affine variety with the Zariski topology (cf. also Affine scheme).

Affine spaces associated with a vector space over a skew-field are constructed in a similar manner.