# lattice in ${\mathbb{R}}^{n}$

###### Definition.

A lattice in ${\mathrm{R}}^{n}$ is an $n$-dimensional additive free group^{} over $\mathrm{Z}$ which generates ${\mathrm{R}}^{n}$ over $\mathrm{R}$.

Example: The following is an example of a lattice $\mathcal{L}\subset {\mathbb{R}}^{2}$, generated by ${w}_{1}=(1,2),{w}_{2}=(4,1)$.

$$\mathcal{L}=\{\alpha {w}_{1}+\beta {w}_{2}\mid \alpha ,\beta \in \mathbb{Z}\}$$ |

Title | lattice in ${\mathbb{R}}^{n}$ |
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Canonical name | LatticeInmathbbRn |

Date of creation | 2013-03-22 13:52:14 |

Last modified on | 2013-03-22 13:52:14 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 7 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 11H06 |

Synonym | lattice |

Synonym | grid |

Related topic | MinkowskisTheorem |

Related topic | PicksTheorem |

Related topic | ProductOfPosets |

Defines | lattice in ${\mathbb{R}}^{n}$ |