Difference between revisions of "Vector module"
From Online Dictionary of Crystallography
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A ''vector module'' is the set of vectors spanned by a number ''n'' of basis vectors | 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, | with integer coefficients. The basis vectors should be independent over the integers, | ||
| − | which means that any linear combination <math>\sum_i m_i | + | which means that any linear combination <math>\sum_i m_i a_i</math> with ''m''<sub>i</sub> integers |
is equal to zero if, and only if, all coefficients ''m''<sub>i</sub> are zero. The term Z-module | is equal to zero if, and only if, all coefficients ''m''<sub>i</sub> are zero. The term Z-module | ||
is sometimes used to underline the condition that the coefficients are integers. | is sometimes used to underline the condition that the coefficients are integers. | ||
Revision as of 19:04, 18 May 2009
Module vectoriel (Fr.)
Synonymous: Z-module
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 [math]\sum_i m_i a_i[/math] 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 n.