December 9th, 2016, 11:57 AM  #1 
Newbie Joined: Dec 2016 From: Natal  Brazil Posts: 10 Thanks: 0  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?

December 9th, 2016, 02:34 PM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
$\displaystyle \lambda _1 v_1 + \lambda _2 v_2 + \lambda _3 v_3 = 0$ 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  
December 9th, 2016, 08:29 PM  #3 
Newbie Joined: Dec 2016 From: Natal  Brazil Posts: 10 Thanks: 0 
Thank you so much !!!

December 10th, 2016, 06:52 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,821 Thanks: 750 
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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dependence, independence, linear 
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