Introduction to a Gram-Schmidt-type biorthogonalization method

Abstract

The aim of this expository/pedagogical paper is to describe a Gram-Schmidt biorthogonalization method in such a way that it can be used as an introduction to the subject for undergraduate presentation. The task of biorthogonalization naturally arises when the scalar product of vectors formed are linear combinations of two sets of linearly independent vectors, as the case may be. If one wants the scalar product to have the usual form, the two sets of basis vectors should be biorthogonal. If they are not, the question of biorthogonalization arises. New is the detailed description of the biorthogonalzation method for teaching purposes as well as the comparison of this method with Schmidt's orthogonalization method in the case when the sets of linearly independent vectors are identical.