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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
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".
This definition of codimension in terms of the number of functions needed to cut out a subspace extends to situations in which both the ambient space and subspace are infinite dimensional. In other language, which is basic for any kind of intersection theory, we are taking the union of a certain number of constraints. We have two phenomena to ...
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.
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 ...
A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.. Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces).
For example, the set of all vectors (x, y, z) (over real or rational numbers) satisfying the equations + + = + = is a one-dimensional subspace. More generally, that is to say that given a set of n independent functions, the dimension of the subspace in K k will be the dimension of the null set of A , the composite matrix of the n functions.