Editing Orthogonalization (section)

{{more citations needed|date=January 2021}}

In [[linear algebra]], '''orthogonalization''' is the process of finding a [[Set (mathematics)|set]] of [[orthogonal vector]]s that [[span (linear algebra)|span]] a particular [[linear subspace|subspace]].  Formally, starting with a [[linearly independent]] set of vectors {''v''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;''v''<sub>''k''</sub>} in an [[inner product space]] (most commonly the [[Euclidean space]] '''R'''<sup>''n''</sup>), orthogonalization results in a set of [[Orthogonality|orthogonal]] vectors {''u''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;''u''<sub>''k''</sub>} that [[Generator (mathematics)|generate]] the same subspace as the vectors ''v''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;''v''<sub>''k''</sub>.  Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same [[linear span]].

In addition, if we want the resulting vectors to all be [[unit vector]]s, then we [[Unit vector|normalize]] each vector and the procedure is called '''orthonormalization'''.

Orthogonalization is also possible with respect to any [[symmetric bilinear form]] (not necessarily an inner product, not necessarily over [[real number]]s), but standard algorithms may encounter [[division by zero]] in this more general setting.