Difference between revisions of "Vector space"
From Online Dictionary of Crystallography
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| − | <font color="blue">Espace vectoriel</font> (''Fr'') | + | <font color="blue">Espace vectoriel</font> (''Fr''). <Font color="black">Spazio vettoriale</Font> (''It''). <Font color="purple">ベクトル空間</Font> (''Ja''). |
| − | For each pair of points X and Y in [[point space]] one can draw a vector '''r''' from X to Y. The set of all vectors forms a '''vector space'''. The vector space has no origin but instead there is the ''zero vector'' which is obtained by connecting any point X with itself. The vector '''r''' has a ''length'' which is designed by |'''r'''| = r, where r is a non–negative real number. This number is also called the ''absolute value'' of the vector. | + | |
| + | For each pair of points ''X'' and ''Y'' in [[point space]] one can draw a vector '''r''' from ''X'' to ''Y''. The set of all vectors forms a '''vector space'''. The vector space has no origin but instead there is the ''zero vector'' which is obtained by connecting any point ''X'' with itself. The vector '''r''' has a ''length'' which is designed by |'''r'''| = ''r'', where ''r'' is a non–negative real number. This number is also called the ''absolute value'' of the vector. | ||
The maximal number of linearly independent vectors in a vector space is called the ''dimension of the space''. | The maximal number of linearly independent vectors in a vector space is called the ''dimension of the space''. | ||
| − | An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if a different origin in point space is chosen. The coordinates of the points change when moving from | + | An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if a different origin in point space is chosen. The coordinates of the points change when moving from one origin to another one. However, the coefficients of the vector '''r''' do not change. |
The [[point space]] is a dual of the vector space because to each vector in vector space a pair of points in [[point space]] can be associated. | The [[point space]] is a dual of the vector space because to each vector in vector space a pair of points in [[point space]] can be associated. | ||
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==See also== | ==See also== | ||
| − | * | + | * Chapter 1.3.2 of ''International Tables for Crystallography, Volume A'', 6th edition |
| − | * [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/22/index.html Matrices, Mappings and Crystallographic Symmetry] | + | * [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/22/index.html ''Matrices, Mappings and Crystallographic Symmetry''] (Teaching Pamphlet No. 22 of the International Union of Crystallography) |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 18:14, 17 May 2017
Espace vectoriel (Fr). Spazio vettoriale (It). ベクトル空間 (Ja).
For each pair of points X and Y in point space one can draw a vector r from X to Y. The set of all vectors forms a vector space. The vector space has no origin but instead there is the zero vector which is obtained by connecting any point X with itself. The vector r has a length which is designed by |r| = r, where r is a non–negative real number. This number is also called the absolute value of the vector.
The maximal number of linearly independent vectors in a vector space is called the dimension of the space.
An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if a different origin in point space is chosen. The coordinates of the points change when moving from one origin to another one. However, the coefficients of the vector r do not change.
The point space is a dual of the vector space because to each vector in vector space a pair of points in point space can be associated.
Face normals, translation vectors, Patterson vectors and reciprocal lattice vectors are elements of vector space.
See also
- Chapter 1.3.2 of International Tables for Crystallography, Volume A, 6th edition
- Matrices, Mappings and Crystallographic Symmetry (Teaching Pamphlet No. 22 of the International Union of Crystallography)