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Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue. The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An ...
Two space groups, considered as subgroups of the group of affine transformations of space, have the same space group type if they are the same up to an affine transformation of space that preserves orientation. Thus e.g. a change of angle between translation vectors does not affect the space group type if it does not add or remove any symmetry.
In probability theory, a simplex space is often used to represent the space of probability distributions. The Dirichlet distribution, for instance, is defined on a simplex. In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc.
Jouanolou's original statement was: If X is a scheme quasi-projective over an affine scheme, then there exists a vector bundle E over X and an affine E-torsor W.. By the definition of a torsor, W comes with a surjective map to X and is Zariski-locally on X an affine space bundle.
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
A function : is called affine if it preserves affine combinations. So (+ +) = + + ()for any affine combination + + in A. The space of affine functions A* is a linear space. The dual vector space of A* is naturally isomorphic to an (n+1)-dimensional vector space F(A) which is the free vector space on A modulo the relation that affine combination in A agrees with affine combination in F(A).
An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All n-dimensional affine spaces over a given field are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space.
The aforementioned facts do not depend on the structure being that of Euclidean space (namely, involving Euclidean distance) and are correct in any affine space. In a Euclidean space: There is the distance between a flat and a point. (See for example Distance from a point to a plane and Distance from a point to a line.)