Ortonormalização duma base(Gram-Schmidt)

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 à 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