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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. Physical space (in pre-relativistic conceptions) is not only an affine space, but it also has a metric structure and in particular a conformal structure. 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! 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. Another way of looking at this is that by forgetting about the zero vector, we're remembering the vector's tail and where it is situated. Denoting the position of the vector's tail by O, the process of adding vectors a and b involves taking the displacements of a and b relative to their tails and applying them sequentially starting from the tail, O. The resulting sum is O + (a - O) + (b - O) = a - O + b. The operation that emerges from this is the ternary affine operation a - b + c. Precise definition An affine space is a set with a faithful freely transitive vector space action, a principal homogeneous space with a vector space action. Alternatively an affine space is a set S, together with a vector space V, and a map such that 1. for every b in S the map is a bijection, and 2. for every a, b and c in S we have Consequences We can define addition of vectors and points as follows 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 is a real number, then is independent of O. Instead of arbitrary linear combinations, only such affine combinations of points have meaning. 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 is an affine space, where 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 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 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 nor of angle. 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. Affine spaces over general fields An affine space over a field, other than the trivial field and the field (0,1,2}, 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 The case of the field may be dealt with separately, and the resolution for the case of the field may be captured by modifying identity (3) to The following properties may be derived 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), 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), take r = 1/(1-t) and apply axiom (3). For (10), the case s = 0 follows from (5). Otherwise, one may take r = 1/(s(1-ms)) and t = ms and apply axiom (3), and then property (5). 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 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. From (6), we get (rs)A = O,rs,A = O,r,O,s,A = r(sA). From (1), we have 0A = O,0,A = O. From (2), we have 1A = O,1,A = A. From (7), we have mB,n,C = mB,n,mC. Using these results, we may employ (9) to show that B/(1-t),t,C/t = 1/(t(1-s)) t(1-s)B/(1-t),t,(1-s)C = 1/(t(1-s)) (t B,s,(1-s)C/s) = 1/(1-s) B,s,(1-s)C/s = B/(1-s),s,C/s for s and t other than 0 and 1. Commutativity of addition then follows from (5) with A+B=A/t,t,B/(1-t) = B/(1-t),1-t,A/t = B+A. Multiplication of the O vector yields O since rO = O,r,O = O, by (4). The additive identity property then follows with O+A=O/(1-t),t,A/t = O,t,A/t = t(1/t)A = 1A = A. Distributivity over vector addition follows with r(A+B)=rA/(1-t),t,B/t = rA/(1-t),t,rB/t = rA+rB. Distributivity over scalar addition follows from (8) with rA+sA = O,r/(1-t),A,t,O,s/t,A = O,r+s,A = (r+s)A. Associativity follows from (10), with A+(B+C) = A/(1-t),t,B/(t(1-s)),s,C/(ts) = C/st,1-st,B/(t(1-s),(1-t)/(1-s),A/(1-t) = C/st,1-st,(A+B)/(1-st) = C+(A+B) = (A+B)+C. This requires s and t to be chosen such that neither is 0 nor 1 and such that st is not 1. Finally, the identification of this operator is established with (1-r)A+rB = (1-r)/(1-r),A,r,rB/r = A,r,B. The cases r = 0 and r = 1 are handled separately using (0) and (1). Affine spaces over the 3-element field The one loose end in the proof of associativity is for the 3-element field, . The only definition for addition available is A+B = 2A,2,B. To establish associativity, one needs to show that A,2,B,2,2C = 2A,2,B,2,C. A systematic exploration of all the combinations of axiom (3) shows that for the 3-element field, the following identities will hold: Writing the operation A,2,B more compactly as AB, the required properties may be more succinctly stated as Instead, one may take as the defining postulates Affine spaces over the 2-element field and "affine groups" The other loose end is in the definition of addition, which breaks down for the field . Since a vector space over does not have any non-trivial multiplication by scalars, it may be equivalently characterized as an abelian group with A+A = 1A+1A = (1+1)A = 0A = 0. A suitable choice for an operation is the ternary operator ABC = A+B+C, for which one may pose the following properties Under (G3), one also has commutativity See also | |||||||
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