# Dual basis

### From Online Dictionary of Crystallography

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# Dual basis

### Other languages

Base duale (*Fr*).

## Definition

The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows:

Given a basis of *n* vectors **e _{i}** spanning the direct space

*E*, and a vector

^{n}**x**=

*x*

^{i}**e**, let us consider the

_{i}*n*quantities defined by the scalar products of

**x**with the basis vectors,

**e**:

_{i}*x _{i}* =

**x**.

**e**=

_{i}*x*

^{j}**e**.

_{j}**e**=

_{i}*x*,

^{j}g_{ji}where the *g _{ji}* 's are the doubly covariant components of the metric tensor.

By solving these equations in terms of *x ^{j}*, one gets:

*x^j^* = *x,,i,, g^ij^*

where the matrix of the *g^ij^* 's is inverse of that of the *g,,ij,,* 's (*g^ik^g,,jk,,* = δ^i^,,j,,). The development of vector **x** with respect to basis vectors **e,,i,,** can now also be written:

x=x^i^e,,i,,=x,,i,, g^ij^e,,j,,

The set of *n* vectors **e^i^** = *g^ij^* **e,,j,,** that span the space *E^n^* forms a basis since vector **x** can be written:

x=x,,i,,e^i^

This basis is the *dual basis* and the *n* quantities *x,,i,,* defined above are the
coordinates of **x** with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:

e,,i,,=g,,ij,,e^j^

The scalar products of the basis vectors of the dual and direct bases are:

g^i^,,j,,=e^i^.e,,j,,=g^ik^e,,k,,.e,,j,,=g^ik^g,,jk,,= δ^i^,,j,,.

One has therefore, since the matrices *g^ik^* and *g,,ij,,* are inverse:

g^i^,,j,,=e^i^.e,,j,,= δ^i^,,j,,.

These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of ["reciprocal space"] are identical.

## Change of basis

In a change of basis where the direct basis vectors and coordinates transform like:

e'=_{j}A_{j}^{i}e;_{i}x'=^{j}B_{i</sub?j}x^i^,

where *A _{j}^{i}* and

*B*are transformation matrices, transpose of one another, the dual basis vectors

_{i}^{j}**e**and the coordinates

^{i}*x*transform according to:

_{i}e'=^{j}B_{i}^j^e;^{i}x'=_{j}A._{j}^{i}x_{i}

The coordinates of a vector in reciprocal space are therefore *covariant* and the dual basis vectors (or reciprocal vectors) *contravariant*.

### See also

metric tensor

reciprocal space

The Reciprocal Lattice (Teaching Pamphlet of the *International Union of Crystallography*)

Section 1.1.2 of *International Tables of Crystallography, Volume D*