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In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame.
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 ...
Affine geometry can be developed in two ways that are essentially equivalent. [3] In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear algebra.
Every real or complex affine or projective space is also a topological space. An affine space is a non-compact manifold; a projective space is a compact manifold. In a real projective space a straight line is homeomorphic to a circle, therefore compact, in contrast to a straight line in a linear of affine space.
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.
An inner product of a real vector space is a positive definite bilinear form, and so characterized by a positive definite quadratic form. A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite).
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines.
It is a Euclidean space and a real affine space, and every Euclidean or affine space is isomorphic to it. It is an analytic manifold , and can be considered as the prototype of all manifolds , as, by definition, a manifold is, near each point, isomorphic to an open subset of R n .