# Vector space

### From Online Dictionary of Crystallography

Espace vectoriel (*Fr*). Vektorraum (*Ge*). Spazio vettoriale (*It*). ベクトル空間 (*Ja*). Espacio vectorial (*Sp*).

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)