Question

DETERMINAR O VETOR COORDENADA [V]s, COM V= (1,2,-1) SENDO S= [(1,1,1), (0,1,1), (0,0,1) BASE DO IR³

Answer 1

$[\vec{v}]_{s}=\left[\begin{array}{c}\alpha_{1}\\\alpha_{2}\\...\\\alpha_{n}\end{array}\right]$

Onde os alfas são os coeficientes da combinação linear dos vetores de s que gera v:

$\boxed{\boxed{\vec{v}=\sum\limits_{i=1}^{n}\alpha_{i}\vec{s_{i}}}}$
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Sendo S uma base do lR³, sabemos que S gera o vetor v, então

$\alpha_{1}(1,1,1)+\alpha_{2}(0,1,1)+\alpha_{3}(0,0,1)=(1,2,-1)$

Encontramos os alfas resolvendo um sistema linear.

Escalonando o sistema por matrizes:

$\left[\begin{array}{cccc}1&0&0&~~~~1\\1&1&0&~~~~2\\1&1&1&~-1\end{array}\right]l_{2}\leftarrow l_{2}-l_{1},~~~l_{3}\leftarrow l_{3}-l_{1}\\\\\\\left[\begin{array}{cccc}1&0&0&~~~~1\\0&1&0&~~~~1\\0&1&1&~-2\end{array}\right]l_{3}\leftarrow l_{3}-l_{2}\\\\\\\left[\begin{array}{cccc}1&0&0&~~~~1\\0&1&0&~~~~1\\0&0&1&~-3\end{array}\right]$

Então:

$\alpha_{1}=1\\\alpha_{2}=1\\\alpha_{3}=-3$

Logo:

$[\vec{v}]_{s}=\left[\begin{array}{c}\alpha_{1}\\\alpha_{2}\\\alpha_{3}\end{array}\right]\\\\\\\boxed{\boxed{[\vec{v}]_{s}=\left[\begin{array}{c}~~1\\~~1\\-3\end{array}\right]}}$