(New page: == The Gram-Schmidt Algorithm == ---- In the simplest review, the Gram-Schmidt Algorithm is shown in the following pattern for vectors '''u''' and '''v'''. : <math> \begin{align} \mathbf...) |
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:<math>\mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} </math>. | :<math>\mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} </math>. | ||
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| + | Ryan Jason Tedjasukmana | ||
| + | ---- | ||
| + | [[Inner_Products_MA265F10Walther|Back to Inner Product Spaces and Orthogonal Complements]] | ||
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| + | [[2010_Fall_MA_265_Walther|Back to MA265 Fall 2010 Prof Walther]] | ||
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| + | [[MA265|Back to MA265]] | ||
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| + | [[Category:MA265Fall2010Walther]] | ||
| + | [[Category:MA265]] | ||
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
