# Linear independence

#### mosvas

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
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

2 people

#### mosvas

Thank you so much !!!

#### Country Boy

Math Team
Notice that topsquark said "If the determinant of the coefficient matrix is not 0 then the system is linearly independent." You want linearly dependent.

1 person