Linear independence

Dec 2016
10
0
Natal - Brazil
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?
 

topsquark

Math Team
May 2013
2,450
1,016
The Astral plane
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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Dec 2016
10
0
Natal - Brazil
Thank you so much !!!
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
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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