6.4: Dual basis
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 n , and a vector x = x i e i , let us consider the 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.