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Encyclopedia > Affine space

In mathematics, an affine space is an abstract structure that generalises the affine-geometric 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, since there is no origin. One-dimensional affine space is the affine line. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


Physical space (in pre-relativistic conceptions) is not only an affine space. It also has a metric structure and in particular a conformal structure.

Contents


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 + (ap) + (bp). 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: In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... John Carlos Baez (b. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

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

The proof is a routine exercise. 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. In mathematics, an affine combination of vectors x1, ..., xn is a linear combination in which the sum of the coefficients is 1, thus: . Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients are scalars in K. This concept is important...


Precise definition

An affine space is a set with a faithful transitive vector space action, a principal homogeneous space with a vector space action. In mathematics, a symmetry group describes all symmetries of objects. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ...


Alternatively an affine space is a set S, together with a vector space V, and a map

Theta : S^2 to V : (a, b) mapsto Theta(a, b) =: a - b,

such that

1. for every b in S the map
Theta_b : S to V : a mapsto a - b,
is a bijection, and
2. for every a, b and c in S we have
(a-b) + (b-c) = a-c.,

Consequences

We can define addition of vectors and points as follows

Phi : S times V to S : (a, v) mapsto a + v := Theta_a^{-1}v.

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


Conversely, any vector space V is an affine space for vector subtraction.


If O, a and b are points in S and ell is a real number, then

oplus_O : S^2 to S : (a, b) mapsto a oplus_O 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. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, an affine combination of vectors x1, ..., xn is a linear combination in which the sum of the coefficients is 1, thus: . Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients are scalars in K. This concept is important...


Affine subspaces

An affine subspace of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set In mathematics, an affine combination of vectors x1, ..., xn is a linear combination in which the sum of the coefficients is 1, thus: . Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients are scalars in K. This concept is important...

S=left {left. sum^N_i alpha_i mathbf{v}_i rightvert sum^N_ialpha_i=1right}

is an affine space, where {vi}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 The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...

W=left{left. sum^N_i beta_imathbf{v}_i rightvert sum^N_i beta_i=0right}.

This vector subspace, and therefore also the affine subspace, is of dimension N–1. This affine subspace can be equivalently described as the coset of the W-action

S=mathbf{p}+W,

where p is any element of S.


One might like to define an affine subspace of an affine space as a set closed under affine combinations. However, affine combinations are only defined in vector spaces; one cannot add points of an affine space. Allowing a slightly more abstract definition, one may define an affine subspace of an affine space as a subset which is left invariant under an affine transformation.


In affine geometry there is not only no notion of origin, but neither a notion of length or angle. In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ...


An affine transformation between two vector spaces is a combination of a linear transformation and a translation. For specifying one the origins are used, but the set of affine transformations does not depend on the origins. In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as...


Intrinsic Definition of Affine Spaces

An affine space over a field, other than the trivial field {0,1}, may be characterized as an algebra equipped with ternary operation [A,r,B] (intuitively to be thought of as (1-r)A + rB) such that

  • (1) [A,0,B] = A
  • (2) [A,1,B] = B
  • (3) [A,rt(1-t),[B,s,C]] = [[A,rt(1-s),B],t,[A,rs(1-t),C]].

The following properties may be derived

  • (4) [A,t,A] = A
  • (5) [A,m,B] = [B,1-m,A]
  • (6) [A,m,[A,n,C]] = [A,mn,C]
  • (7) [A,m,[B,n,C]] = [[A,m,B],n,[A,m,C]]
  • (8) [[A,m,B],t,[A,n,B]] = [A,m(1-t)+nt,B]
  • (9) [A,m,[B,1/m,C]] = [B,2-m,[A,1/(2-m),C]] for m other than 0 or 2.
  • (10) [A,1/(2-s),[B,s,C]] = [B,1/(2-s),[A,s,C]] for s other than 0 or 2.
  • (11) [A,t,[B,s,C]] = [[A,t(1-s)/(1-t),B],t,[A,s,C]] for t other than 1.

Property (4) is derived by taking r = 0 in axiom (3) and applying axiom (1).


For (5), the case m = 1 trivial. For m other than 1, we may set r = 1/(1-m), s = 0 and t = 1-m and apply axiom (3),

  • [A,m,B] = [A,m,[B,0,C]] = [[A,1,B],1-m,[A,0,C]] = [B,1-m,A]

For (6), the case m = 1 is also trivial. In other cases, we may take r=n/(1-m), s = 1, t = m in axiom (3), which leads directly to the result.


For (7), the cases n = 0 or 1 are trivial. In other cases, we may write r = m/(n(1-n)) and s = t = n, in axiom (3) to directly arrive at the result.


For (8), the cases t = 0 or 1 are trivial. Otherwise, if m(1-t)+nt = 0, one use property (5) to rewrite this as [[B,1-n,A],1-t,[B,1-m,A]] = [B,(1-n)(1-(1-t))+(1-m)(1-t),A] and prove property (8) for this, instead. Otherwise, we may take r = ((1-t)m+tn)/(t(1-t)), s = nt/((1-t)m+nt) and derive the result directly from axiom (3).


For (9), the case m = 1 is trivial. In other cases, we may take r = m((2-m)(m-1)), s = 1/m and t = 2-m and apply axiom (3).


For (10), again s = 1 is easily handled. For other s, take r = (2-s)/(1-s), t = 1/(2-s) and apply axiom (3).


For (11), take r = 1/(1-t) and apply axiom (3).


With these properties in hand, we may show that a vector space may be defined by first selecting a point O to designate as the zero vector and then defining the operations

  • rA = [O,r,A]
  • A+B = [A/(1-t),t,B/t] for any t other than 0 or 1

The second operation may then be proven to be independent of t; the properties of a vector space may be derived and one may prove that with these definitions that [A,r,B] reduces to (1-r)A+rB.


(To be continued)


See also


  Results from FactBites:
 
Affine geometry - Wikipedia, the free encyclopedia (756 words)
Affine geometry can be explained as the geometry of vectors, not involving any notions of length or angle.
Affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0.
The term affine space is used in projective geometry as the complement of the points (hyperplane) at infinity (see also projective space).
Affine space - Wikipedia, the free encyclopedia (1025 words)
In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.
An affine space is a set with a faithful transitive vector space action, a principal homogeneous space with a vector space action.
An affine subspace of a vector space V is a subset closed under affine combinations of vectors in the space.
  More results at FactBites »

 
 

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