Orthonormalizing a Given Basis

Introduction

Abstract   We introduce the Gram-Schmidt procedure which turns any basis of a subspace \(V\) of \(\RNrSpc{n}\) into a orthonormal basis of \(V\).

Outline   In the previous sections we saw how convenient it can be to work with a basis of orthonormal vectors vs. an arbitrary basis of a vector space \(V\). It is therefore useful to know that an arbitrary ordered basis can always be transformed into an orthonormal one. The procedure which accomplishes this feat is called Gram-Schmidt orthonormalization, named after Jorgen Pedersen Gram and Eduard Schmidt. It had, however, been used earlier in specialized contexts. So, here goes:

If \(\mathcal{A}=(\Vect{a}_1,\dots ,\Vect{a}_n)\) is an arbitrary ordered set of linearly independent vectors in \(\RNrSpc{n}\), let \(V\) denote its span. The first step toward transforming \(\mathcal{A}\) into an ONB of \(V\) is just normalizing \(\Vect{a}_1\):

The second step is to split \(\Vect{a}_2 = \Vect{b}_2 + \Vect{u}_2\), with \(\Vect{b}_2\in \SpanOfSet{ \Vect{v}_1 }\) and \(\Vect{u}_2 \in \SpanOfSet{ \Vect{v}_1 }^{\bot}\). Then \(\Vect{b}_2\) contributes nothing new to \(\SpanOfSet{\Vect{v}_1}\); so forget about it. However, \(\Vect{v}_1\), and \(\Vect{v}_2\DefEq \frac{\Vect{u}_2}{ \Norm{\Vect{u}_2} }\) are orthonormal vectors by design, and they satisfy

Next we split \(\Vect{a}_3 = \Vect{b}_3 + \Vect{u}_3\), with \(\Vect{b}_3\in \SpanOfSet{ \Vect{v}_1,\Vect{v}_2 }\) and \(\Vect{u}_3 \in \SpanOfSet{ \Vect{v}_1,\Vect{v}_2 }^{\bot}\). Then \(\Vect{b}_3\) contributes nothing new to \(\SpanOfSet{\Vect{v}_1, \Vect{v}_2}\); so forget about it. However, \(\Vect{v}_1,\Vect{v}_2\), and \(\Vect{v}_3\DefEq \frac{\Vect{u}_3}{ \Norm{\Vect{u}_3} }\) are orthonormal vectors by design, and they satisfy

We continue in this manner until we have processed all the vectors in \(\mathcal{A}\), and the result is an ONB \(\Vect{v}_1,\dots ,\Vect{v}_r\) for \(V\). Of course, at some point we must check that this procedure actually accomplishes what it claims to do.

As a consequence of the Gram-Schmidt orthonormalization procedure, we see that every orthonormal set in a subspace \(V\) or \(\RNrSpc{n}\) can be complemented to an ONB of \(V\).