Diferenças entre edições de "Ortogonalização e normalização"

Edição atual desde as 13h15min 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: easy
TEMPO MEDIO DE RESOLUCAO: 10 mn
TEMPO MAXIMO DE RESOLUCAO: 30 mn
PALAVRAS CHAVE:

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

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-Considere a seguinte base de \( \mathbb{R³} \) \(\left\{\left(\begin{array}{c}-3\\3\\-1\\\end{array}\right),\left(\begin{array}{c}-2\\-3\\0\\\end{array}\right),\left(\begin{array}{c}2\\2\\2\\\end{array}\right)\right\}\). Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.
+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)\(\left\{\left(\begin{array}{c}-\frac{3}{\sqrt{19}}\\\frac{3}{\sqrt{19}}\\-\frac{1}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{47}{\sqrt{4522}}\\-24\sqrt{\frac{2}{2261}}\\-\frac{3}{\sqrt{4522}}\\\end{array}\right),\left(\begin{array}{c}-\frac{3}{\sqrt{238}}\\\sqrt{\frac{2}{119}}\\\frac{15}{\sqrt{238}}\\\end{array}\right)\right\}\),
+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)\(\left\{\left(\begin{array}{c}-\frac{2}{\sqrt{13}}\\-\frac{3}{\sqrt{13}}\\0\\\end{array}\right),\left(\begin{array}{c}-\frac{45}{\sqrt{3094}}\\15\sqrt{\frac{2}{1547}}\\-\sqrt{\frac{13}{238}}\\\end{array}\right),\left(\begin{array}{c}-\frac{3}{\sqrt{238}}\\\sqrt{\frac{2}{119}}\\\frac{15}{\sqrt{238}}\\\end{array}\right)\right\}\),
+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)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\end{array}\right),\left(\begin{array}{c}-2\sqrt{\frac{2}{21}}\\\frac{5}{\sqrt{42}}\\-\frac{1}{\sqrt{42}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{2}{7}}\\-\frac{1}{\sqrt{14}}\\\frac{3}{\sqrt{14}}\\\end{array}\right)\right\}\),
+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)\(\left\{\left(\begin{array}{c}-\frac{3}{\sqrt{19}}\\\frac{3}{\sqrt{19}}\\-\frac{1}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}4\sqrt{\frac{2}{133}}\\\frac{11}{\sqrt{266}}\\\frac{9}{\sqrt{266}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{2}{7}}\\-\frac{1}{\sqrt{14}}\\\frac{3}{\sqrt{14}}\\\end{array}\right)\right\}\)
+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[]
+Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/1695923671436179/instanciasGramSchmidt.zip]
 Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

Diferenças entre edições de "Ortogonalização e normalização"

Edição atual desde as 13h15min de 28 de julho de 2016

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