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Affine SpaceThe view of physical space in classical mechanics is that of an affine space. All points are equal and from one point you can get to any other by a suitable vector. Informal description An affine space is a space in which you can subtract two points to form a vector pointing from one point to the other. If you single out one point and identify it with the zero vector you get a vector space. Since in any vector space you can subtract vectors to get a connecting vector, all vector spaces are affine spaces. Another way of putting this is that an affine space is a vector space that's forgotten its origin. Instead of arbitrary linear combinations, only affine combinations of points have meaning. Formal description In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. Definition An affine space is a set with a transitive vector space action. Alternatively an affine space is a set S, together with a vector space V, and a map such that - for every a in S the map is a bijection,
- for every a, b and c in S we have .
We can define addition of vectors and points as follows Examples - If V be a vector space, then V is an affine space for vector subtraction.
- If O, a and b are points in S and l is a real number, then is independend of O.
See also: affine geometry.
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