International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 5

## Section 1.1.2.1. Change of basis

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France

#### 1.1.2.1. Change of basis

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Let us consider a vector space spanned by the set of n basis vectors , , . The decomposition of a vector using this basis is written using the Einstein convention. The interpretation of the position of the indices is given below. For the present, we shall use the simple rules:

 (i) the index is a subscript when attached to basis vectors; (ii) the index is a superscript when attached to the components. The components are numerical coordinates and are therefore dimensionless numbers.

Let us now consider a second basis, . The vector x is independent of the choice of basis and it can be decomposed also in the second basis:

If and are the transformation matrices between the bases and , the following relations hold between the two bases: (summations over j and i, respectively). The matrices and are inverse matrices: (Kronecker symbol: if if ).

Important Remark. The behaviour of the basis vectors and of the components of the vectors in a transformation are different. The roles of the matrices and are opposite in each case. The components are said to be contravariant. Everything that transforms like a basis vector is covariant and is characterized by an inferior index. Everything that transforms like a component is contravariant and is characterized by a superior index. The property describing the way a mathematical body transforms under a change of basis is called variance.