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Consider the vectors v = < 1 , 3 > w = < 3 , 2 >, and x = < 1 , 0 >. If the vectors v, w and x are linearly independent, answer with 0. If they aren't, find coefficients a,b and c, not all 0, such that 

a < 1 , 3 > + b < 3 , 2 >  + c < 1 , 0 > = < 0 , 0 > and answer with {a+b}/c. 

 May 19, 2019
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3 vectors having two coordinates each cannot possibly be linearlly independent.

 

\(\text{we have the equation}\\ \begin{pmatrix}1&3&1\\3&2&0\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\)

 

\(\text{apply Gaussian elimination}\\ \begin{pmatrix}1&3&1\\3&2&0\end{pmatrix}\\ \begin{pmatrix}1&3&1\\0&-7&-3\end{pmatrix}\\ \begin{pmatrix}1&3&1\\0&1&\frac 3 7\end{pmatrix}\\ \begin{pmatrix}1&0&-\frac 2 7\\0&1&\frac 3 7\end{pmatrix}\\ \)

 

\(a-\dfrac 2 7 c = 0,~c=\dfrac 7 2 a\\ b + \dfrac 3 7 c = 0,~c = -\dfrac 7 3 b\\ b = -\dfrac 3 2 a\\ \left(a,b,c\right)= \left(1,-\dfrac 3 2,\dfrac 7 2\right)\)

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 May 19, 2019
edited by Rom  May 20, 2019

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