The Gram Matrix is also known as the Gramian matrix.
The Gram matrix of a set of vectors is the symmetric matrix of inner products.
The entries are given by Gij = viT vj = (vi | vj).
|viT vj =||
A set of vectors is linearly independent if and only if: det(Gij) ≠ 0
Gram Matrix as the covariance matrix
if ei represents the basis vectors then:
e*i = Gij ei
represents the reciprocal or dual basis.
If the basis is orthogonal then Gij = δij = identity matrix and so the the basis and its dual are effectively equal.