# general linear group

Given a vector space^{} $V$, the general linear group^{} $\mathrm{GL}(V)$ is defined to be the group of invertible linear transformations from $V$ to $V$. The group operation^{} is defined by composition: given $T:V\u27f6V$ and ${T}^{\prime}:V\u27f6V$ in $\mathrm{GL}(V)$, the product^{} $T{T}^{\prime}$ is just the composition of the maps $T$ and ${T}^{\prime}$.

If $V={\mathbb{F}}^{n}$ for some field $\mathbb{F}$, then the group $\mathrm{GL}(V)$ is often denoted $\mathrm{GL}(n,\mathbb{F})$ or ${\mathrm{GL}}_{n}(\mathbb{F})$. In this case, if one identifies each linear transformation $T:V\u27f6V$ with its matrix with respect to the standard basis, the group $\mathrm{GL}(n,\mathbb{F})$ becomes the group of invertible^{} $n\times n$ matrices with entries in $\mathbb{F}$, under the group operation of matrix multiplication^{}.

One also discusses the general linear group on a module $M$ over some ring $R$. There it is the set of automorphisms^{} of $M$ as an $R$-module. For example, one might take $\mathrm{GL}(\mathbb{Z}\oplus \mathbb{Z})$; this is isomorphic to the group of two-by-two matrices with integer entries having determinant^{} $\pm 1$. If $M$ is a general $R$-module, there need not be a natural interpretation^{} of $\mathrm{GL}(M)$ as a matrix group.

The general linear group is an example of a group scheme; viewing it in this way ties together the properties of $\mathrm{GL}(V)$ for different vector spaces $V$ and different fields $F$. The general linear group is an algebraic group, and it is a Lie group if $V$ is a real or complex vector space.

When $V$ is a finite-dimensional Banach space^{}, $\mathrm{GL}(V)$ has a natural topology coming from the operator norm^{}; this is isomorphic to the topology^{} coming from its embedding^{} into the ring of matrices. When $V$ is an infinite-dimensional vector space, some elements of $\mathrm{GL}(V)$ may not be continuous^{} and one generally looks instead at the set of bounded operators^{}.

Title | general linear group |
---|---|

Canonical name | GeneralLinearGroup |

Date of creation | 2013-03-22 12:25:36 |

Last modified on | 2013-03-22 12:25:36 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20G15 |

Related topic | Group |

Related topic | Representation |

Related topic | SpecialLinearGroup |