# Base ortonormal para um subespaço de $$R^3$$

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• 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
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)$$\left\{\left(\begin{array}{c}\frac{1}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\-\frac{3}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{21}{\sqrt{646}}\\3\sqrt{\frac{2}{323}}\\-\frac{13}{\sqrt{646}}\\\end{array}\right),\left(\begin{array}{c}-\frac{2}{\sqrt{17}}\\\frac{3}{\sqrt{17}}\\\frac{2}{\sqrt{17}}\\\end{array}\right)\right\}$$.