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The group is an example of a unipotent linear algebraic group, the group is an example of a solvable algebraic group called the Borel subgroup of (). It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of G L ( n ) {\displaystyle \mathrm {GL} (n)} is conjugated into B {\displaystyle B} .
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.
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.
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
Linear algebra is the branch of mathematics concerning linear equations such as: + ... making possible the general linear group.
The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices , where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose ).
Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
The cyclic group C 3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.. In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself.
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