Ortogonalização e normalização - Histórico de revisões

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Ist178052: Criou a página com "
'''Metadata'''
CONTEXTO : Primeiro ciclo universitário AREA:..."

2016-07-28T08:08:18Z

Criou a página com "<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"> '''Metadata''' <div class="mw-collapsible-content"> *CONTEXTO : Primeiro ciclo universitário *AREA:..."

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<div class="toccolours mw-collapsible mw-collapsed" style="width:420px">
'''Metadata'''
<div class="mw-collapsible-content">
*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:
</div>
</div>

Para obter o zip que contém as instâncias deste exercício clique aqui[]

Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

← Revisão anterior		Revisão das 12h15min de 28 de julho de 2016
Linha 24:		Linha 24:


−	Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/~~1695923671436177~~/instanciasGramSchmidt.zip]	+	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		Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

@@ Linha 16: / Linha 16: @@
 </div>
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-Considere a seguinte base de \( \mathbb{R³} \) \(\left\{\left(\begin{array}{c}-1\\-2\\-1\\\end{array}\right),\left(\begin{array}{c}-1\\1\\2\\\end{array}\right),\left(\begin{array}{c}-1\\-2\\1\\\end{array}\right)\right\}\). Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.
+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)\(\left\{\left(\begin{array}{c}-\frac{1}{\sqrt{6}}\\-\sqrt{\frac{2}{3}}\\-\frac{1}{\sqrt{6}}\\\end{array}\right),\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\0\\\frac{1}{\sqrt{2}}\\\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{3}}\\-\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\end{array}\right)\right\}\),
@@ Linha 24: / Linha 24: @@
-Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/1695923671436176/instanciasGramSchmidt.zip]
+Para obter o zip que contém as instâncias deste exercício clique aqui[https://drive.tecnico.ulisboa.pt/download/1695923671436177/instanciasGramSchmidt.zip]
 Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

← Revisão anterior		Revisão das 08h38min de 28 de julho de 2016
Linha 24:		Linha 24:


−	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/1695923671436176/instanciasGramSchmidt.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

@@ Linha 16: / Linha 16: @@
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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 \( \mathbb{R³} \) \(\left\{\left(\begin{array}{c}-1\\-2\\-1\\\end{array}\right),\left(\begin{array}{c}-1\\1\\2\\\end{array}\right),\left(\begin{array}{c}-1\\-2\\1\\\end{array}\right)\right\}\). 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)\(\left\{\left(\begin{array}{c}-\frac{1}{\sqrt{6}}\\-\sqrt{\frac{2}{3}}\\\frac{1}{\sqrt{6}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{5}{42}}\\4\sqrt{\frac{2}{105}}\\\frac{11}{\sqrt{210}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{5}{7}}\\\frac{1}{\sqrt{35}}\\-\frac{3}{\sqrt{35}}\\\end{array}\right)\right\}\)

@@ Linha 16: / Linha 16: @@
 </div>
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-Considere a seguinte base de \( \mathbb{R³}\) \(\left\{\left(\begin{array}{c}3\\1\\-1\\\end{array}\right),\left(\begin{array}{c}1\\0\\0\\\end{array}\right),\left(\begin{array}{c}-2\\-1\\0\\\end{array}\right)\right\}\)Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.
+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.
-A)\(\left\{\left(\begin{array}{c}\frac{3}{\sqrt{11}}\\\frac{1}{\sqrt{11}}\\-\frac{1}{\sqrt{11}}\\\end{array}\right),\left(\begin{array}{c}\sqrt{\frac{2}{11}}\\-\frac{3}{\sqrt{22}}\\\frac{3}{\sqrt{22}}\\\end{array}\right),\left(\begin{array}{c}0\\-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}\\\end{array}\right)\right\}\),
+A)\(\left\{\left(\begin{array}{c}-\frac{3}{\sqrt{19}}\\\frac{3}{\sqrt{19}}\\-\frac{1}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}-\frac{47}{\sqrt{4522}}\\-24\sqrt{\frac{2}{2261}}\\-\frac{3}{\sqrt{4522}}\\\end{array}\right),\left(\begin{array}{c}-\frac{3}{\sqrt{238}}\\\sqrt{\frac{2}{119}}\\\frac{15}{\sqrt{238}}\\\end{array}\right)\right\}\),
-B)\(\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}&#038;\frac{1}{\sqrt{3}}&#038;-\frac{1}{\sqrt{3}}\\-\frac{7}{\sqrt{78}}&#038;\sqrt{\frac{2}{39}}&#038;-\frac{5}{\sqrt{78}}\\\frac{1}{\sqrt{26}}&#038;-2\sqrt{\frac{2}{13}}&#038;-\frac{3}{\sqrt{26}}\\\end{array}\right)\),
+B)\(\left\{\left(\begin{array}{c}-\frac{2}{\sqrt{13}}\\-\frac{3}{\sqrt{13}}\\0\\\end{array}\right),\left(\begin{array}{c}-\frac{45}{\sqrt{3094}}\\15\sqrt{\frac{2}{1547}}\\-\sqrt{\frac{13}{238}}\\\end{array}\right),\left(\begin{array}{c}-\frac{3}{\sqrt{238}}\\\sqrt{\frac{2}{119}}\\\frac{15}{\sqrt{238}}\\\end{array}\right)\right\}\),
-C)\(\left(\begin{array}{ccc}-\frac{2}{3}&#038;\frac{1}{3}&#038;-\frac{2}{3}\\\frac{11}{3\sqrt{26}}&#038;\frac{4\sqrt{\frac{2}{13}}}{3}&#038;-\frac{7}{3\sqrt{26}}\\\frac{1}{\sqrt{26}}&#038;-2\sqrt{\frac{2}{13}}&#038;-\frac{3}{\sqrt{26}}\\\end{array}\right)\),
+C)\(\left\{\left(\begin{array}{c}\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\frac{1}{\sqrt{3}}\\\end{array}\right),\left(\begin{array}{c}-2\sqrt{\frac{2}{21}}\\\frac{5}{\sqrt{42}}\\-\frac{1}{\sqrt{42}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{2}{7}}\\-\frac{1}{\sqrt{14}}\\\frac{3}{\sqrt{14}}\\\end{array}\right)\right\}\),
-D)\(\left(\begin{array}{ccc}0&#038;-\frac{1}{\sqrt{2}}&#038;-\frac{1}{\sqrt{2}}\\-2\sqrt{\frac{2}{17}}&#038;\frac{3}{\sqrt{34}}&#038;-\frac{3}{\sqrt{34}}\\\frac{3}{\sqrt{17}}&#038;\frac{2}{\sqrt{17}}&#038;-\frac{2}{\sqrt{17}}\\\end{array}\right)\)
+D)\(\left\{\left(\begin{array}{c}-\frac{3}{\sqrt{19}}\\\frac{3}{\sqrt{19}}\\-\frac{1}{\sqrt{19}}\\\end{array}\right),\left(\begin{array}{c}4\sqrt{\frac{2}{133}}\\\frac{11}{\sqrt{266}}\\\frac{9}{\sqrt{266}}\\\end{array}\right),\left(\begin{array}{c}-\sqrt{\frac{2}{7}}\\-\frac{1}{\sqrt{14}}\\\frac{3}{\sqrt{14}}\\\end{array}\right)\right\}\)

@@ Linha 16: / Linha 16: @@
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-Considere a seguinte base de \( \mathbb{R³} \)\(\left(\begin{array}{ccc}1&#038;1&#038;-1\\0&#038;-1&#038;-1\\-2&#038;1&#038;-2\\\end{array}\right)\)Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.
+Considere a seguinte base de \( \mathbb{R³} \(\left\{\left(\begin{array}{c}3\\1\\-1\\\end{array}\right),\left(\begin{array}{c}1\\0\\0\\\end{array}\right),\left(\begin{array}{c}-2\\-1\\0\\\end{array}\right)\right\}\)Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.
 A)\(\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}&#038;\frac{1}{\sqrt{3}}&#038;-\frac{1}{\sqrt{3}}\\0&#038;-\frac{1}{\sqrt{2}}&#038;-\frac{1}{\sqrt{2}}\\-\sqrt{\frac{2}{3}}&#038;\frac{1}{\sqrt{6}}&#038;-\frac{1}{\sqrt{6}}\\\end{array}\right)\),

← Revisão anterior		Revisão das 08h09min de 28 de julho de 2016
Linha 16:		Linha 16:
	</div>		</div>
	</div>		</div>
		+	Considere a seguinte base de \( \mathbb{R³} \)\(\left(\begin{array}{ccc}1&1&-1\\0&-1&-1\\-2&1&-2\\\end{array}\right)\)Diga qual dos seguintes conjuntos corresponde á ortonormalização desta base.

		+	A)\(\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}\\0&-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{6}}\\\end{array}\right)\),
		+	B)\(\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}\\-\frac{7}{\sqrt{78}}&\sqrt{\frac{2}{39}}&-\frac{5}{\sqrt{78}}\\\frac{1}{\sqrt{26}}&-2\sqrt{\frac{2}{13}}&-\frac{3}{\sqrt{26}}\\\end{array}\right)\),
		+	C)\(\left(\begin{array}{ccc}-\frac{2}{3}&\frac{1}{3}&-\frac{2}{3}\\\frac{11}{3\sqrt{26}}&\frac{4\sqrt{\frac{2}{13}}}{3}&-\frac{7}{3\sqrt{26}}\\\frac{1}{\sqrt{26}}&-2\sqrt{\frac{2}{13}}&-\frac{3}{\sqrt{26}}\\\end{array}\right)\),
		+	D)\(\left(\begin{array}{ccc}0&-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-2\sqrt{\frac{2}{17}}&\frac{3}{\sqrt{34}}&-\frac{3}{\sqrt{34}}\\\frac{3}{\sqrt{17}}&\frac{2}{\sqrt{17}}&-\frac{2}{\sqrt{17}}\\\end{array}\right)\)