Difference between revisions of "Vector space"
From Online Dictionary of Crystallography
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| − | <font color="blue">Espace vectoriel</font> (''Fr''); <Font color="black">Spazio vettoriale</Font>(''It''); | + | <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. | ||
Revision as of 12:12, 20 December 2016
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 an origin to the other 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 8.1 in the International Tables for Crystallography Volume A
- Matrices, Mappings and Crystallographic Symmetry, teaching pamphlet No. 22 of the International Union of Crystallography