# Vector module

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 |

with *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 the reciprocal space, it is sometimes called *Fourier module*.