| Line 3: | Line 3: | ||
| − | In the simplest review, the Gram-Schmidt Algorithm is shown in the following pattern for vectors '''u | + | In the simplest review, the Gram-Schmidt Algorithm is shown in the following pattern for the given vectors '''u'''. |
: <math> | : <math> | ||
\begin{align} | \begin{align} | ||
Latest revision as of 15:38, 6 December 2010
The Gram-Schmidt Algorithm
In the simplest review, the Gram-Schmidt Algorithm is shown in the following pattern for the given vectors u.
- $ \begin{align} \mathbf{u}_1 & = \mathbf{v}_1, \\ \mathbf{u}_2 & = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_2), \\ \mathbf{u}_3 & = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_3)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_3), \\ & {}\ \ \vdots \\ \mathbf{u}_k & = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,(\mathbf{v}_k), \end{align} $
where
- $ \mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} $.
Ryan Jason Tedjasukmana
Back to Inner Product Spaces and Orthogonal Complements
