Affine space

The affine space, sometimes also called linear manifold, takes in the systematic development of geometry a middle position between Euclidean space and projective space.

The affine space in the narrowest sense is a mathematical model for us familiar three-dimensional space of intuition.

In a broader sense, an affine space, like other mathematical spaces also have an arbitrary dimension: as affine space can also be a single point, the affine line, the affine plane and four - and higher-dimensional spaces, respectively. In general, these areas are only finite-dimensional.

Various mathematical disciplines have found different clarifications of the term.

The affine space in linear algebra

Definition

Given a set whose elements are geometrically considered as points, a vector space over a field and a mapping from to, which assigns two points a connection vector, so that the following two rules:

  • For every three points applies: (triangle rule, the relationship of Chasles )
  • For each point and each vector there exists a unique point such that ( Abtragbarkeitsregel ).

The triple is called affine space. If it is clear which vector space and which arrow figure is based, one also speaks solely on affine space A.

Affine subspace

In affine space as a representation of an addition is defined by that which is just about uniquely determined point Q. For fixed, the corresponding figure is called translation ( shift) or precise translation by the vector and is then called the associated translation vector.

Translations are always bijections. They form together with the series connection as a group linking a subgroup of the symmetric group by, with and for always and apply.

If P is a fixed point of A and U a given subspace of, then is also an affine subspace of A. Instead of the term " affine subspace " is often the equivalent term used affine subspace. The belonging to an affine subspace subspace is uniquely determined by.

The dimension of an affine space A to a vector space of a body is defined as the dimension of the vector space over. Often it is convenient to consider also the empty set as an affine (partial ) space. This empty subspace dimension of -1 is then assigned.

The affine point space and its associated vector space

For these reasons, it is sometimes omitted rigid distinction between the affinity space of a point part, and the displacement vectors of the vector space of the other.

Examples

  • The -dimensional Euclidean space is the affine space over a - dimensional Euclidean vector space (ie one - dimensional vector space with scalar product ).
  • The solutions of an inhomogeneous linear system of equations form an affine space over the vector space of solutions of the corresponding homogeneous system. This similarly applies to systems of linear differential equations.
  • In the differential geometry of affine spaces play a role in the theory of fiber bundles. Examples of the fibers of the affine tangent, the connection beam and Jetbündeln.

Use in algebraic geometry

  • In classical algebraic geometry of the n-dimensional affine space An over an algebraically closed field k is the algebraic variety kn.
  • In modern algebraic geometry of the n-dimensional affine space AnA is defined over a commutative ring A with identity as the spectrum of the polynomial ring A [ X1, ..., Xn ] in n indeterminates. For an A- algebra B, the B- valued points of AnA are equal to Bn.

Definitions of synthetic geometry

An affine space in the sense of synthetic geometry consists of the following data:

  • A set of points
  • A lot of straight lines
  • An incidence relation that specifies lie which points on which line
  • A parallel relation indicating which lines are parallel,

So that certain axioms are satisfied, which suggests the view, including Euclid's famous parallel postulate.

The structures defined in this way generalize the notion of affine space, which is defined in this article. Thus, the following applies:

→ See the article mentioned, in which the generalized structures are described for more details. As the term " affine space " ( as a room with shifts that form a vector space ) can be distinguished from the axiomatic concepts of synthetic geometry in the article Affine geometry is shown in more detail.

Mathematics Null vector Spectrum of a ring Algebra over a field Affine plane (incidence geometry)
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