Affine General Linear Group
In mathematics, the affine group or general affine group of any affine space over a field (mathematics), field is the group (mathematics), group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relation to general linear group Construction from general linear group Concretely, given a vector space , it has an underlying affine space obtained by "forgetting" the origin, with acting by translations, and the affine group of can be described concretely as the semidirect product of by , the general linear group of : :\operatorname(V) = V \rtimes \operatorname(V) The action of on is the natural one (linear transformations are automorphisms), so this defines a semidirect product. In terms of matrices, one writes: :\operatorname(n,K) = K^n \rtimes \operatorname(n,K) where here the natural action of on is matrix multiplication of a vector. Stabilizer of a point Given the affine group of an af ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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