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In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point that serves as an origin.

1 Informal descriptions
2 Precise definition
3 Examples
4 Affine subspaces
5 Affine combinations and affine dependence
6 Axioms
7 See also
8 References

Informal descriptions

The following characterization may be easier to understand than a precise definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of mathematical physicist John Baez, "An affine space is a vector space that's forgotten its origin"). Imagine that Smith knows that a certain point is the true origin, and Jones believes that another point — call it p — is the origin. Two vectors, a and b, are to be added. Jones draws an arrow from p to a and another arrow from p to b, and completes the parallelogram to find what Jones thinks is a + b, but Smith knows that it is actually p + (a − p) + (b − p). Similarly, Jones and Smith may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However — and note this well:

If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!

Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space.

Precise definition

An affine space can most easily be defined in terms of a vector space over a field (or division ring), as here, but can also be defined intrinsically by axioms, without reference to an auxiliary vector space or field.

An affine space is a set with a faithful freely transitive vector space action, i.e. a torsor (or principal homogeneous space) for the vector space.

Equivalently an affine space is a set A, together with a vector space V, and a map

$V \times A\to A$ , written as (v, a) → v + a.

The map has the properties that:

1. for every a in A the map $V \to A\colon v \mapsto v + a\,$ is a bijection, and

2. for every v, w in V and a in A we have $(v+w)+a = v+(w+a).\,$

We can define subtraction of points of an affine space as follows:

a − b is the unique vector in V such that (a − b) + b = a.

By choosing an origin a we can thus identify A with V, hence change A into a vector space.

Conversely, any vector space V is an affine space over itself.

If O, a and b are points in A and $\ell$ is a scalar, then

$\ell a+(1-\ell)b = O+\ell(a-O)+(1-\ell)(b-O)\,$

is independent of O. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

Examples

Any coset of a subspace V of a vector space is an affine space over V.
If A is a matrix and b lies in its column space, the set of solutions of the equation Ax = b is an affine space over the subspace of solutions of Ax = 0.
The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation.
More generally, if $T\colon V\to W$ is a linear mapping and y lies in its image, the set of solutions $x\in V$ to the equation T(x) = y is a coset of the kernel of T, and is therefore an affine space over Ker(T).

Affine subspaces

An affine subspace (sometimes called a linear manifold) of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set

$A=\Bigl\{\sum^N_i \alpha_i \mathbf{v}_i \Big| \sum^N_i\alpha_i=1\Bigr\}$

is an affine space, where {v_i}_i is a family of vectors in V. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace W of V

$W=\Bigl\{\sum^N_i \beta_i\mathbf{v}_i \Big| \sum^N_i \beta_i=0\Bigr\}.$

This affine subspace can be equivalently described as the coset of the W-action

$S=\mathbf{p}+W,\,$

where p is any element of A. A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.

For example, in R³, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

Affine combinations and affine dependence

An affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three. Vectors

v₁, v₂, ..., v_n

are linearly dependent if there exist scalars a₁, a₂, … ,a_n, not all zero, for which

a₁v₁ + a₂v₂ + … + a_nv_n = 0

(1)

Similarly they are affinely dependent if the same is true and also

$\sum_{i=1}^n a_i = 0.$

Equation (1) is an affine relation among the vectors v₁, v₂, …, v_n.

Axioms

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.

Coxeter (1969, p.192) axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.

Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint):

Any two distinct points lie on a unique line.
Given a point and line there is a unique line which contains the point and is parallel to the line
There exist three non-collinear points.

As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a "line at infinity" whose points correspond to equivalence classes of parallel lines.

(Cameron 1991, chapter 3) gives axioms for higher dimensional affine spaces.

References

Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes, 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019, http://www.maths.qmul.ac.uk/~pjc/pps/
Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, MR 123930, ISBN 978-0-471-50458-0
Dolgachev, I.V.; Shirokov, A.P. (2001), "Affine space", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/A/a011100.htm
Ernst Snapper and Robert J. Troyer, Metric Affine Geometry, Dover Publications; Reprint edition (October 1989)

Categories: Affine geometry | Linear algebra

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