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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 ...
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
In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space. [1] Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.
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".
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 ...
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 ...
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...
In mathematics, the affine hull or affine span of a set S in Euclidean space R n is the smallest affine set containing S, [1] or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace. The affine hull aff(S) of S is the set of all affine combinations of elements ...