Affine space

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Affine space

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In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. 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.

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[edit] 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, importantly if the sum of the coefficients in a linear combination is 1, then Smith and Jones will arrive at the same answer.

Extending our previous example: Smith, knowing the true origin, can deduce that the point that Jones thinks is λa + (1 − λ)b is actually p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b. Since λ + (1 − λ) = 1, Smith and Jones describe the same point with the same linear combination in their respective frames of reference.

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.

[edit] 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 point 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 point 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.

[edit] 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).

[edit] Affine subspaces

An affine subspace (sometimes called a linear manifold or linear variety) 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 {vi}i is a family of vectors in V – this space is the affine span of these point. 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, or equivalently as any level set of the quotient map V \to V/W. A choice of p gives a base point of A and an identification of W with A, but there is no natural choice, nor a natural identification of W with 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.

[edit] 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, …, vn

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

a1v1 + a2v2 + … + anvn = 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, …, vn.

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

[edit] Relation to projective spaces

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An affine space is a subspace of projective space, which is in turn a quotient of a vector space.

Affine spaces are subspaces of projective spaces: 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 as a closure by adding a "line at infinity" whose points correspond to equivalence classes of parallel lines.

Further, transformations of projective space that preserve affine space (equivalently, that preserve the points at infinity as a set) yield transformations of affine space, and conversely any affine linear transformation extends uniquely to a projective linear transformations, so affine transformations are a subset of projective transforms. Most familiar is that Möbius transformations (transformations of the projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.

However, one cannot take the projectivization of an affine space, so projective spaces are not naturally quotients of affine spaces: one can only take the projectivization of a vector space, since the projective space is lines through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.

[edit] See also

[edit] References

Retrieved from "http://en.wikipedia.org/wiki/Affine_space"

Categories: Affine geometry | Linear algebra


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