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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
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 group GL n (K) itself; 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.
A representation of a group on an -dimensional real vector space is simply a group homomorphism : (,) from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.
In mathematics, the special linear group SL(2, R) or SL 2 (R) is the group of 2 × 2 real matrices with determinant one: (,) = {():,,, =}.It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics.
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication.The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special [1] automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. [2]
A fundamental example of a reductive group is the general linear group of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group G m is the group GL (1), and so its group G m ( k ) of k -rational points is the group k * of nonzero elements of k under multiplication.
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