# Dual basis

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 ei spanning the direct space En, and a vector x = xi ei, let us consider the n quantities defined by the scalar products of x with the basis vectors, ei:

xi = x . ei = xj ej . ei = xj gji,

where the gji 's are the doubly covariant components of the metric tensor.

By solving these equations in terms of xj, 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 = Aji ei; x'j = Bi</sub?j x^i^,
```

where Aji and Bij are transformation matrices, transpose of one another, the dual basis vectors ei and the coordinates xi transform according to:

```e'j  = Bi^j^ ei; x'j = Ajixi.
```

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