Vector module

(Difference between revisions)
 Revision as of 16:50, 8 July 2017 (view source)m (small correction to sentence structure)← Older edit Latest revision as of 14:42, 20 November 2017 (view source) (Added German translation (U. Mueller)) Line 1: Line 1: - Module vectoriel (''Fr''). Modulo vettoriale (''It''). - Synonyms: Z-module, Fourier module Synonyms: Z-module, Fourier module + + + Module vectoriel (''Fr''). Vektormodul (''Ge''). Modulo vettoriale (''It''). + == Definition == == Definition ==

Latest revision as of 14:42, 20 November 2017

Synonyms: Z-module, Fourier module

Module vectoriel (Fr). Vektormodul (Ge). Modulo vettoriale (It).

Definition

A vector module is the set of vectors spanned by a number n of basis vectors with integer coefficients. The basis vectors should be independent over the integers, which means that any linear combination
 ∑ miai i
with mi integers is equal to zero if, and only if, all coefficients mi are zero. The term Z-module is sometimes used to underline the condition that the coefficients are integers. The number of basis vectors is the rank of the vector module.

Comment

An n-dimensional lattice in an n-dimensional vector space is an example of a vector module, with rank n. In reciprocal space, the reciprocal lattice corresponding to a crystallographic structure is a special case of a vector module. The Bragg peaks for the crystal fall on the positions of the reciprocal lattice. More generally, the Bragg peaks of an m-dimensional aperiodic crystal structure belong to a vector module of rank n, larger than m. To indicate that this module exists in reciprocal space, it is sometimes called the Fourier module.