Editing Gram–Schmidt process (section)

== Expressed using geometric algebra ==
Expressed using notation used in [[geometric algebra]], the unnormalized results of the Gram–Schmidt process can be expressed as
<math display="block">\mathbf{u}_k = \mathbf{v}_k - \sum_{j=1}^{k-1} (\mathbf{v}_k \cdot \mathbf{u}_j)\mathbf{u}_j^{-1}\ ,</math>
which is equivalent to the expression using the <math>\operatorname{proj}</math> operator defined above. The results can equivalently be expressed as<ref>{{cite book |last1=Doran |first1=Chris |last2=Lasenby |first2=Anthony |title=Geometric Algebra for Physicists |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-71595-9 |page=124 }}</ref>
<math display="block">\mathbf{u}_k = \mathbf{v}_{k}\wedge\mathbf{v}_{k-1}\wedge\cdot\cdot\cdot\wedge\mathbf{v}_{1}(\mathbf{v}_{k-1}\wedge\cdot\cdot\cdot\wedge\mathbf{v}_{1})^{-1},</math>
which is closely related to the expression using determinants above.