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A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class.
The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix. It is denoted by either GL( F ) or GL(∞, F ) , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.
Linear (pronounced: / l ɪ ˈ n ɪər /; lin-EER [1]) is an American freestyle-pop group from Fort Lauderdale, Florida. [2]The lineup originally consisted of founder, vocalist, and main songwriter Charlie Pennachio, bassist and guitar player Wyatt Pauley, Vocalist, Rap and percussion player Joey Restivo, drummer Gerald Rappaport, guitarist Phil Conneilly, and backup vocalist/keyboardist Ricki ...
In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation.
Generalized orthogonal group, generalized special orthogonal group. The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group of hyperbolic motions of the Poincaré disk model of the Hyperbolic plane. Lorentz group; Spinor group; Symplectic group; Exceptional groups G 2; F 4; E 6; E 7; E 8; Affine group ...
Special linear group; T. Thin group (algebraic group theory) U. Unit hyperbola; V. Virasoro group
For a linear algebraic group G over the real numbers R, the group of real points G(R) is a Lie group, essentially because real polynomials, which describe the multiplication on G, are smooth functions. Likewise, for a linear algebraic group G over C, G(C) is a complex Lie group. Much of the theory of algebraic groups was developed by analogy ...