# Vector module

### From Online Dictionary of Crystallography

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Synonyms: Z-module, Fourier module | Synonyms: Z-module, Fourier module | ||

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+ | <font color="blue">Module vectoriel</font> (''Fr''). <font color="red">Vektormodul</font> (''Ge''). <font color="black">Modulo vettoriale</font> (''It''). | ||

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== 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

∑ | m_{i}a_{i} |

i |

*m*

_{i}integers is equal to zero if, and only if, all coefficients

*m*

_{i}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*.