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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
The special linear group SL n (K) (the subgroup of matrices with determinant 1); The group of invertible upper (or lower) triangular matrices; If g i is a collection of elements in GL n (K) indexed by a set I, then the subgroup generated by the g i is a linear group.
The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings. The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group, the general linear group of degree n, denoted GL n (R).
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.
group operation is addition N 0 0 abelian C n: n: C ×: nonzero complex numbers with multiplication N 0 Z: abelian C: 1 GL(n,C) general linear group: invertible n×n complex matrices: N 0 Z: For n=1: isomorphic to C ×: M(n,C) n 2: SL(n,C) special linear group: complex matrices with determinant. 1 N 0 0 for n=1 this is a single point and thus ...
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
Let be the general linear group GL n of invertible matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group W {\displaystyle W} is isomorphic to the symmetric group S n {\displaystyle S_{n}} on n {\displaystyle n} letters, with permutation matrices as representatives.
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group , defined by the relation M T M = I n {\displaystyle M^{T}M=I_{n}} where M T {\displaystyle M^{T}} is the transpose of M {\displaystyle M} .