# special linear group

Given a vector space^{} $V$, the special linear group^{} $\mathrm{SL}(V)$ is defined to be the subgroup^{} of the general linear group^{} $\mathrm{GL}(V)$ consisting of all invertible linear transformations $T:V\u27f6V$ in $\mathrm{GL}(V)$ that have determinant^{} 1.

If $V={\mathbb{F}}^{n}$ for some field $\mathbb{F}$, then the group $\mathrm{SL}(V)$ is often denoted $\mathrm{SL}(n,\mathbb{F})$ or ${\mathrm{SL}}_{n}(\mathbb{F})$, and if one identifies each linear transformation with its matrix with respect to the standard basis, then $\mathrm{SL}(n,\mathbb{F})$ consists of all $n\times n$ matrices with entries in $\mathbb{F}$ that have determinant 1.

Title | special linear group |
---|---|

Canonical name | SpecialLinearGroup |

Date of creation | 2013-03-22 12:25:38 |

Last modified on | 2013-03-22 12:25:38 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20G15 |

Synonym | SL |

Related topic | GeneralLinearGroup |

Related topic | Group |

Related topic | UnimodularMatrix |