Converting a General Basis to an Orthonormal Basis

A non-orthonormal basis can be made orthonormal using the Gram-Schmidt Procedure.

The Gram-Schmidt procedure works by the properties of orthogonal projection:

Gram-Schmidt Procedure

Let $\{{\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_k\}$ be a basis for the inner product space, $V$.

1. ${\hat{u}}_1=\frac{1}{\Vert {\vec{v}}_1\Vert }{\vec{v}}_1$
2. ${\hat{u}}_2$
3. ${\vec{u}}_2={\vec{v}}_2-{\text{proj}}_{{\hat{u}}_1}({\vec{v}}_2)$
4. ${\hat{u}}_2=\frac{1}{\Vert {\vec{u}}_2\Vert }{\vec{u}}_2$
5. ${\hat{u}}_3$
6. ${\vec{u}}_3={\vec{v}}_3-{\text{proj}}_{{\hat{u}}_1}({\vec{v}}_3)-{\text{proj}}_{{\hat{u}}_2}({\vec{v}}_3)$
7. ${\hat{u}}_3$ = $\frac{1}{\Vert {\vec{u}}_3\Vert }{\vec{u}}_3$ …
8. ${\hat{u}}_k$
9. ${\vec{u}}_k={\vec{v}}_k-{\text{proj}}_{{\hat{u}}_1}({\vec{v}}_k)-\cdots -{\text{proj}}_{{\hat{u}}_{k-1}}({\vec{v}}_k)$
10. ${\hat{u}}_k=\frac{1}{{\vec{u}}_k}{\vec{u}}_k$

#todo Write as an algorithm?