Difference between revisions of "Vector module"
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| − | + | Synonyms: Z-module, Fourier module | |
| − | < | + | <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 == | |
| − | 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, 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 sometimes used to underline the condition that the coefficients are integers. The number of basis vectors is the ''rank'' of the vector module. |
| − | 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 ''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 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 | An ''n''-dimensional lattice in an ''n''-dimensional vector space is an example | ||
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The Bragg peaks for the crystal fall on the positions of the reciprocal lattice. | 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 | More generally, the Bragg peaks of an ''m''-dimensional aperiodic crystal structure | ||
| − | belong to a vector module of rank ''n'', larger than '' | + | 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''. | ||
| + | |||
| + | [[Category:Fundamental crystallography]] | ||
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 [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 m. To indicate that this module exists in reciprocal space, it is sometimes called the Fourier module.