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
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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
.
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
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where
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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.
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207 |
Comments
An affine isomorphism is also called an affine collineation. An equi-affine group is also called a Euclidean group.
References
[a1] | M. Berger, "Geometry" , I , Springer (1987) pp. Chapt. 2 MR0903026 MR0895392 MR0882916 MR0882541 Zbl 0619.53001 Zbl 0606.51001 Zbl 0606.00020 |
Affine space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_space&oldid=45049