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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.
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 ...
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
The pair (P, η) defines the structure of an affine geometry on M, making it into an affine manifold. The affine Lie algebra aff(n) splits as a semidirect product of R n and gl(n) and so η may be written as a pair (θ, ω) where θ takes values in R n and ω takes values in gl(n).
In mathematics, an affine combination of x 1, ..., x n is a linear combination
An affine (algebraic) variety is an affine algebraic set that is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible . If X is an affine algebraic set, and I is the ideal of all polynomials that are zero on X , then the quotient ring R = k [ x 1 , … , x n ] / I {\displaystyle R=k ...
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting" [1] [2]) the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines.