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An affine space is a subspace of a projective space, which is in turn the quotient of a vector space by an equivalence relation (not by a linear subspace) Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on it, and conversely any ...
2 Examples. 3 History. 4 References. ... Download as PDF; Printable version; In other projects ... any convex subset of a real affine space is a convex space.
Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it). This forms a homogeneous (+)-dimensional geometry. An affine space is not modular (for example, if and are parallel lines then the formula in the definition of modularity fails). However, it is easy to check that all localizations are modular.
Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication; Flat (geometry), a Euclidean subspace; Affine subspace, a geometric structure that generalizes the affine properties of a flat; Projective subspace, a geometric structure that generalizes a linear subspace of a vector 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.)
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".
See Affine space § Affine combinations and barycenter for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry , because the set of all affine combinations of a set of points forms the smallest affine space containing the points, exactly as the linear combinations of a set of vectors form their ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...