Linear Algebra with Applications, 10th Edition

5.6 The Gram–Schmidt Orthogonalization Process

In this section, we learn a process for constructing an orthonormal basis for an n-dimensional inner product space V. The method involves using projections to transform an ordinary basis $\{{\mathbf{x}}_1,{\mathbf{x}}_2,\mathrm{\dots },{\mathbf{x}}_n\}$ into an orthonormal basis $\{{\mathbf{u}}_1,{\mathbf{u}}_2,\mathrm{\dots },{\mathbf{u}}_n\}$.

We will construct the ${\mathbf{u}}_i\text{’s}$ so that

$$\text{Span}=({\mathbf{u}}_1,\mathrm{\dots },{\mathbf{u}}_k)=\text{Span}({\mathbf{x}}_1,\mathrm{\dots },{\mathbf{x}}_k)$$

for $k=1,\mathrm{\dots },n$. To begin the process, let

$${\mathbf{u}}_1=\left({\displaystyle \frac{1}{\Vert {\mathbf{x}}_1\Vert }}\right){\mathbf{x}}_1$$ (1)

$\text{Span}({\mathbf{u}}_1)=\text{Span}({\mathbf{x}}_1)$, since ${\mathbf{u}}_1$ is a unit vector in the direction of ${\mathbf{x}}_1$. Let ${\mathbf{p}}_1$ denote the projection of ${\mathbf{x}}_2$ onto $\text{Span}({\mathbf{x}}_1)=\text{Span}({\mathbf{u}}_1)$; that is,

$${\mathbf{p}}_1=\langle {\mathbf{x}}_2,{\mathbf{u}}_1\rangle {\mathbf{u}}_1$$

By Theorem 5.5.7,

$$({\mathbf{x}}_2-{\mathbf{p}}_1)\perp {\mathbf{u}}_1$$

Note that ${\mathbf{x}}_2-{\mathbf{p}}_1\ne \mathbf{0}$, since

$${\mathbf{x}}_2-{\mathbf{p}}_1=\frac{-\langle {\mathbf{x}}_2,{\mathbf{u}}_1\rangle }{\Vert {\mathbf{x}}_1\Vert }{\mathbf{x}}_1+{\mathbf{x}}_2$$ (2)

and ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ are linearly independent. If we set

$${\mathbf{u}}_2={\displaystyle \frac{1}{\Vert {\mathbf{x}}_2-{\mathbf{p}}_1\Vert }}({\mathbf{x}}_2-{\mathbf{p}}_1)$$ (3)

then ${\mathbf{u}}_2$ is a unit vector orthogonal ...