Lineaariseryhmä

Lineaariseryhmä, or linear group, is a fundamental concept in abstract algebra, a branch of mathematics. It refers to a group (a set equipped with a binary operation satisfying certain conditions) where the group operation is addition, and the set is a vector space over a field. The most common examples of linear groups are the general linear group GL(n, F), the special linear group SL(n, F), and the orthogonal group O(n, F), where n is a positive integer and F is a field.

The general linear group GL(n, F) consists of all invertible matrices of size n × n with entries in F. The special linear group SL(n, F) is the subgroup of GL(n, F) consisting of matrices with determinant 1. The orthogonal group O(n, F) is the subgroup of GL(n, F) consisting of matrices whose transpose is equal to their inverse.

Linear groups are important in various areas of mathematics, including representation theory, algebraic geometry, and number theory. They also have applications in physics, particularly in the context of symmetry groups in quantum mechanics.

The study of linear groups often involves the classification of these groups, the determination of their properties, and the investigation of their representations. This can be a complex task, as the number of linear groups can be very large, even for relatively small values of n and F.