The dual basis is a basis associated to the basis of a
Given a basis of n vectors ei spanning the direct space En, and a vector x = x i ei, let us consider the n quantities defined by the
xi = x . ei = x j ej . ei = x j gji,
where the gji 's are the doubly covariant components of the metric tensor.
x j = xi gij
where the matrix of the gij 's is inverse of that of the gij 's (gikgjk = δij). The
x = x i ei = xi gij ej
The set of n vectors ei = gij ej that span the space En forms a basis since vector x can be written:
x = xi ei
This basis is the dual basis and the n quantities xi 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:
ei = gij ej
The scalar products of the basis vectors of the dual and direct bases are:
gij = ei . ej = gik ek . ej = gikgjk = δij.
One has therefore, since the matrices gik and gij are inverse:
gij = ei . ej = δij.
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 j xi,
where Aji and Bi j 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.
Section 1.1.2 of International Tables of Crystallography, Volume D