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="purple">ベクトル空間</Font> (''Ja'').
<font color="blue">Espace vectoriel</font> (''Fr''). <font color="red">Vektorraum</font> (''Ge''). <font color="black">Spazio vettoriale</font> (''It''). <font color="purple">ベクトル空間</font> (''Ja''). <font color="green">Espacio vectorial</font> (''Sp'').

Latest revision as of 14:43, 20 November 2017

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