Diferenças entre edições de "Ortogonalização e normalização em subespaço"

Fonte: My Solutions
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(Há uma edição intermédia do mesmo utilizador que não está a ser apresentada)
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Considere o subespaço \(W= \mathscr{L} \{ \)\(\left(\begin{array}{c}1\\-2\\-2\\\end{array}\right)\)\(\left(\begin{array}{c}-2\\-3\\3\\\end{array}\right)\)\(\left(\begin{array}{c}3\\8\\-4\\\end{array}\right)\)\( \} \). Diga qual dos seguintes conjuntos é uma base ortonormal para \(W\).
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Considere o subespaço \(W= \mathscr{L} \)\(\left\{\left(\begin{array}{c}1\\-3\\-3\\\end{array}\right),\left(\begin{array}{c}-3\\3\\1\\\end{array}\right),\left(\begin{array}{c}-8\\12\\8\\\end{array}\right)\right\}\). Diga qual dos seguintes conjuntos é uma base ortonormal para \(W\).
  
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A)\(\left(\begin{array}{c}\frac{1}{3}\\-\frac{2}{3}\\-\frac{2}{3}\\\end{array}\right)\)\(\left(\begin{array}{c}-\frac{8\sqrt{\frac{2}{97}}}{3}\\-\frac{31}{3\sqrt{194}}\\\frac{23}{3\sqrt{194}}\\\end{array}\right)\),
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A)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right)\right\}\),
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B)\(\left(\begin{array}{c}\frac{1}{3}\\-\frac{2}{3}\\-\frac{2}{3}\\\end{array}\right)\)\(\left(\begin{array}{c}-\frac{8\sqrt{\frac{2}{97}}}{3}\\-\frac{31}{3\sqrt{194}}\\\frac{23}{3\sqrt{194}}\\\end{array}\right)\)\(\left(\begin{array}{c}\frac{3}{\sqrt{89}}\\\frac{8}{\sqrt{89}}\\-\frac{4}{\sqrt{89}}\\\end{array}\right)\),
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B)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{3}{\sqrt{19}}\\\frac{3}{\sqrt{19}}\\\frac{1}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{2}{\sqrt{17}}\\\frac{3}{\sqrt{17}}\\\frac{2}{\sqrt{17}}\\\end{array}\right)\right\}\),
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C)\(\left(\begin{array}{c}\frac{1}{9}\\\frac{4}{9}\\\frac{4}{9}\\\end{array}\right)\)\(\left(\begin{array}{c}\frac{128}{873}\\\frac{961}{1746}\\\frac{529}{1746}\\\end{array}\right)\),
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C)\(\left\{\left(\begin{array}{c}\frac{1}{19}\\\frac{9}{19}\\\frac{9}{19}\\\end{array}\right),\left(\begin{array}{c}\frac{441}{646}\\\frac{18}{323}\\\frac{169}{646}\\\end{array}\right)\right\}\),
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D)\(\left(\begin{array}{c}\frac{1}{3}\\-\frac{2}{3}\\-\frac{2}{3}\\\end{array}\right)\)\(\left(\begin{array}{c}-\sqrt{\frac{2}{11}}\\-\frac{3}{\sqrt{22}}\\\frac{3}{\sqrt{22}}\\\end{array}\right)\)
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D)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right),\left(\begin{array}{c}-\frac{2}{\sqrt{17}}\\\frac{3}{\sqrt{17}}\\\frac{2}{\sqrt{17}}\\\end{array}\right)\right\}\)
  
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Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/1695923671436186/instanciasGramSchmidt2.zip]
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Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/1695923671436185/instanciasGramSchmidt2.zip]
 
  
 
Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt
 
Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

Edição atual desde as 14h58min de 28 de julho de 2016

Metadata

  • CONTEXTO : Primeiro ciclo universitário
  • AREA: Matemática
  • DISCIPLINA: Álgebra Linear
  • ANO: 1
  • LINGUA: pt
  • AUTOR: Equipa Álgebra Linear
  • MATERIA PRINCIPAL: Produtos internos e normas
  • DESCRICAO: Ortogo e norm em subespaço
  • DIFICULDADE: very easy
  • TEMPO MEDIO DE RESOLUCAO: 10 mn
  • TEMPO MAXIMO DE RESOLUCAO: 30 mn
  • PALAVRAS CHAVE:

Considere o subespaço \(W= \mathscr{L} \)\(\left\{\left(\begin{array}{c}1\\-3\\-3\\\end{array}\right),\left(\begin{array}{c}-3\\3\\1\\\end{array}\right),\left(\begin{array}{c}-8\\12\\8\\\end{array}\right)\right\}\). Diga qual dos seguintes conjuntos é uma base ortonormal para \(W\).

A)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right)\right\}\), B)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{3}{\sqrt{19}}\\\frac{3}{\sqrt{19}}\\\frac{1}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{2}{\sqrt{17}}\\\frac{3}{\sqrt{17}}\\\frac{2}{\sqrt{17}}\\\end{array}\right)\right\}\), C)\(\left\{\left(\begin{array}{c}\frac{1}{19}\\\frac{9}{19}\\\frac{9}{19}\\\end{array}\right),\left(\begin{array}{c}\frac{441}{646}\\\frac{18}{323}\\\frac{169}{646}\\\end{array}\right)\right\}\), D)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right),\left(\begin{array}{c}-\frac{2}{\sqrt{17}}\\\frac{3}{\sqrt{17}}\\\frac{2}{\sqrt{17}}\\\end{array}\right)\right\}\)

Para obter o zip que contém as instâncias deste exercício clique aqui[1]

Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

Obtida de "http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Ortogonalização_e_normalização_em_subespaço&oldid=1485"