# Dual basis

The dual basis is a basis associated to the basis of a

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

**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 *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*'s (

_{ij}*g*= δ

^{ik}g_{jk}^{i}

_{j}). The

**x**with

**e**can now also be written:

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

**e**=

_{i}*x*

_{i}g^{ij}**e**

_{j}The set of *n* vectors **e ^{i}** =

*g*

^{ij}**e**that span the space

_{j}*E*forms a basis since vector

^{n}**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*= δ

^{ik}g_{jk}^{i}

_{j}.

One has therefore, since the matrices *g ^{ik}* and

*g*are inverse:

_{ij}*g ^{i}_{j}* =

**e**.

^{i}**e**= δ

_{j}^{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}^{ 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

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