Everything about General Linear Group totally explained
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.
To be more precise, it's necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over
R (the set of
real numbers) is the group of
n×
n invertible matrices of real numbers, and is denoted by
GLn(
R) or
GL(
n,
R).
More generally, the general linear group of degree
n over any
field F (such as the
complex numbers), or a
ring R (such as the ring of
integers), is the set of
n×
n invertible matrices with entries from
F (or
R), again with matrix multiplication as the group operation. Typical notation is
GLn(
F) or
GL(
n,
F), or simply
GL(
n) if the field is understood.
The
special linear group, written
SL(
n,
F) or
SLn(
F), is the
subgroup of
GL(
n,
F) consisting of matrices with
determinant =1.
The group
GL(
n,
F) and its subgroups are often called
linear groups or
matrix groups. These groups are important in the theory of
group representations, and also arise in the study of spatial
symmetries and symmetries of
vector spaces in general, as well as the study of
polynomials. The
modular group may be realised as a quotient of the special linear group SL(2,
Z).
If
n ≥ 2, then the group
GL(
n,
F) isn't
abelian.
General linear group of a vector space
If
V is a
vector space over the field
F, the general linear group of
V, written GL(
V) or Aut(
V), is the group of all
automorphisms of
V, for example the set of all
bijective linear transformations
V →
V, together with functional composition as group operation. If
V has finite
dimension n, then GL(
V) and GL(
n,
F) are
isomorphic. The isomorphism isn't
canonical; it depends on a choice of
basis in
V. Given a basis (
e1, ...,
en) of
V and an automorphism
T in GL(
V), we've
» , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.
It is used in
algebraic K-theory to define
K1, and over the reals has a well-understood topology, thanks to
Bott periodicity.
Further Information
Get more info on 'General Linear Group'.
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