Ortonormalização duma base(Gram-Schmidt)

Metadata

• CONTEXTO : Primeiro ciclo universitário
• AREA: Matemática
• DISCIPLINA: Álgebra Linear
• ANO: 1
• LINGUA: pt
• AUTOR: Ana Moura Santos e Miguel Dziergwa
• MATERIA PRINCIPAL: Produtos internos e normas
• DESCRICAO: Ortonormalização duma base (Gram-Schmidt)
• DIFICULDADE: **
• TEMPO MEDIO DE RESOLUCAO: 10 mn
• TEMPO MAXIMO DE RESOLUCAO: 20 mn
• PALAVRAS CHAVE: base de espaço linear, base ortonormal (ortonormada), vetores ortogonais, vetores unitários, norma 1

Considere a seguinte base de $\mathbb{R}^3$ $\left\{\left(\begin{array}{c}-1\\-2\\-1\\\end{array}\right),\left(\begin{array}{c}-1\\1\\2\\\end{array}\right),\left(\begin{array}{c}-1\\-2\\1\\\end{array}\right)\right\}$. Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.

A) $\left\{\left(\begin{array}{c}-\frac{1}{\sqrt{6}}\\-\sqrt{\frac{2}{3}}\\-\frac{1}{\sqrt{6}}\\\end{array}\right),\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\0\\\frac{1}{\sqrt{2}}\\\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{3}}\\-\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\end{array}\right)\right\}$; B) $\left\{\left(\begin{array}{c}-\frac{1}{\sqrt{6}}\\\frac{1}{\sqrt{6}}\\\sqrt{\frac{2}{3}}\\\end{array}\right),\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}\\0\\\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{3}}\\-\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\end{array}\right)\right\}$; C) $\left\{\left(\begin{array}{c}-\frac{1}{\sqrt{6}}\\-\sqrt{\frac{2}{3}}\\\frac{1}{\sqrt{6}}\\\end{array}\right),\left(\begin{array}{c}-\frac{1}{\sqrt{30}}\\-\sqrt{\frac{2}{15}}\\-\sqrt{\frac{5}{6}}\\\end{array}\right),\left(\begin{array}{c}-\frac{2}{\sqrt{5}}\\\frac{1}{\sqrt{5}}\\0\\\end{array}\right)\right\}$; D) $\left\{\left(\begin{array}{c}-\frac{1}{\sqrt{6}}\\-\sqrt{\frac{2}{3}}\\\frac{1}{\sqrt{6}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{5}{42}}\\4\sqrt{\frac{2}{105}}\\\frac{11}{\sqrt{210}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{5}{7}}\\\frac{1}{\sqrt{35}}\\-\frac{3}{\sqrt{35}}\\\end{array}\right)\right\}$.

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