the transpose and is the volume of the parallelepiped formed by the basis vectors and In general the dual basis of a basis in a finite dimensional vector space can be readily computed as follows: given the basis and corresponding dual basis we can build matrices Then the defining property of the dual basis states that Hence the matrix for the dual basis can be computed as See also Reciprocal lattice Miller index Zone axis Notes References Linear algebra he:מרחב דואלי#הבסיס הדואלי zh:对偶基