10 Orthonormalization

Suppose that we have a set of functions ? k, k=1, 2, …, that are linearly independent but not necessarily orthonormal. We can always find a new set of functions that span the same space and are orthonormal by using the GramSchmidt orthonormalization method. Suppose that the ? k, k=1, 2,…, are not orthonormal. Take ? 1 and form

Then take ? 2 and form

Continuing in the obvious manner, we get {? 'k}, which is an orthonormal set.