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'''Procedura e ortogonalizmit Gram-Shmit''' është një metodë nga algjebra lineare që aplikohet për të marrë një
==Proçeduara Gram–Shmit==
Le te percaktojme nje [[operator]] [[projektimi(algjebra lineare)|projektimi]] te dhene nga
:<math>\mathrm{proj}_{\mathbf{u}}\,\mathbf{v} = {\langle \mathbf{u}, \mathbf{v}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} = {\langle \mathbf{u}, \mathbf{v}\rangle} {\mathbf{u}\over\langle \mathbf{u}, \mathbf{u}\rangle}, </math>
ku <'''u''', '''v'''> japin [[produkti i brendshem|produktin e brendshem]] te vektoreve '''u''' dhe '''v'''. Ky operator projekton vektorin '''v''' ortogonalisht mbi vektor '''u'''.
Procesi Gram–Shmit aplikohet si me poshte:
{|
|width="20px"| ||<math>\mathbf{u}_1 = \mathbf{v}_1,</math>
|width="20px"| ||<math>\mathbf{e}_1 = {\mathbf{u}_1 \over \|\mathbf{u}_1\|}</math>
|-
|| ||<math>\mathbf{u}_2 = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,\mathbf{v}_2, </math>
|| ||<math>\mathbf{e}_2 = {\mathbf{u}_2 \over \|\mathbf{u}_2\|}</math>
|-
|| ||<math>\mathbf{u}_3 = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,\mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_2}\,\mathbf{v}_3, </math>
|| ||<math>\mathbf{e}_3 = {\mathbf{u}_3 \over \|\mathbf{u}_3\|}</math>
|-
|| ||<math>\mathbf{u}_4 = \mathbf{v}_4-\mathrm{proj}_{\mathbf{u}_1}\,\mathbf{v}_4-\mathrm{proj}_{\mathbf{u}_2}\,\mathbf{v}_4-\mathrm{proj}_{\mathbf{u}_3}\,\mathbf{v}_4, </math>
|| ||<math>\mathbf{e}_4 = {\mathbf{u}_4 \over \|\mathbf{u}_4\|}</math>
|-
|| ||align="center"|<math>\vdots</math>
|| ||align="center"|<math>\vdots</math>
|-
|| ||<math>\mathbf{u}_k = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,\mathbf{v}_k, </math>
|| ||<math>\mathbf{e}_k = {\mathbf{u}_k\over \|\mathbf{u}_k \|}</math>
|}
[[Image:Gram–Schmidt process.svg|right|frame|Dy hapat e para te procedures Gram–Schmidt.]]
Sekuenca '''u'''<sub>1</sub>, …, '''u'''<sub>''k''</sub> eshte bashkesia e vektoreve ortogonale. Gjithashtu vektoret e normalizuar '''e'''<sub>1</sub>, …, '''e'''<sub>''k''</sub> formojne nje bashkesi [[ortonormal|ortho''normale'']].
{{mate-cung}}
[[Category:Matematikë]]
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