Diferenças entre edições de "Base ortonormal para um subespaço de \(R^3\)"

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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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Considere o subespaço expansão linear \(W= \mathscr{L} \)\(\left\{\left(\begin{array}{c}1\\-3\\-3\\\end{array}\right),\left(\begin{array}{c}-3\\3\\1\\\end{array}\right),\left(\begin{array}{c}-8\\12\\8\\\end{array}\right)\right\}\). Diga qual dos seguintes conjuntos é uma base ortonormal para \(W\).
  
A)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right)\right\}\),
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A) \(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right)\right\}\);
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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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)\(\left\{\left(\begin{array}{c}\frac{1}{19}\\\frac{9}{19}\\\frac{9}{19}\\\end{array}\right),\left(\begin{array}{c}\frac{441}{646}\\\frac{18}{323}\\\frac{169}{646}\\\end{array}\right)\right\}\),
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C) \(\left\{\left(\begin{array}{c}\frac{1}{19}\\\frac{9}{19}\\\frac{9}{19}\\\end{array}\right),\left(\begin{array}{c}\frac{441}{646}\\\frac{18}{323}\\\frac{169}{646}\\\end{array}\right)\right\}\);
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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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[https://drive.tecnico.ulisboa.pt/download/1695923671436186/instanciasGramSchmidt2.zip]
 
Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/1695923671436186/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

Revisão das 23h49min de 8 de dezembro 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 expansão linear \(W= \mathscr{L} \)\(\left\{\left(\begin{array}{c}1\\-3\\-3\\\end{array}\right),\left(\begin{array}{c}-3\\3\\1\\\end{array}\right),\left(\begin{array}{c}-8\\12\\8\\\end{array}\right)\right\}\). 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