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December 9th, 2016, 11:57 AM   #1
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Unhappy Linear independence

For what real values of x do the vectors v1 = (1, 2, x), v2 = (1, 1, 1) and v3 = (x, 6, 2) form a linearly dependent set?
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December 9th, 2016, 02:34 PM   #2
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Originally Posted by mosvas View Post
For what real values of x do the vectors v1 = (1, 2, x), v2 = (1, 1, 1) and v3 = (x, 6, 2) form a linearly dependent set?
For them to be dependent means that we can find $\displaystyle \lambda _i$ such that
$\displaystyle \lambda _1 v_1 + \lambda _2 v_2 + \lambda _3 v_3 = 0$
where at least one of the $\displaystyle \lambda _i \neq 0$.

The other, more efficient way, is to use matrices:
$\displaystyle \left ( \begin{matrix} 1 & 1 & x \\ 2 & 1 & 6 \\ x & 1 & 2 \end{matrix} \right ) \cdot \left ( \begin{matrix} \lambda _1 \\ \lambda _2 \\ \lambda _3 \end{matrix} \right ) = \left ( \begin{matrix} 0\\ 0 \\ 0 \end{matrix} \right )$

If the determinant of the coefficient matrix is not 0 then the system is linearly independent.

-Dan
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December 9th, 2016, 08:29 PM   #3
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Thank you so much !!!
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December 10th, 2016, 06:52 AM   #4
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Notice that topsquark said "If the determinant of the coefficient matrix is not 0 then the system is linearly independent." You want linearly dependent.
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