Advanced Engineering Mathematics, 7th Edition

7.7 Gram–Schmidt Orthogonalization Process

INTRODUCTION

In Section 7.6 we saw that a vector space V can have many different bases. Recall, the defining characteristics of any basis B = {x₁, x₂, … , x_n} of a vector space V is that

• the set B is linearly independent, and
• the set B spans the space.

In this context the word span means that every vector in the space can be expressed as a linear combination of the vectors x₁, x₂, … , x_n. For example, every vector u in Rⁿ can be written as a linear combination of the vectors in the standard basis B = {e₁, e₂, … , e_n}, where

e₁ = 〈1, 0, 0, … , 0〉, e₂ = 〈0, 1, 0, … , 0〉, …, e_n = 〈0, 0, 0, … , 1〉.

This standard basis B = {e₁, e₂, … , e_n} is also an example of an orthonormal basis; that is, the e