I need to know if these vectors are linearly independent or not- According to answer at least two should be but I don't know how to find out please help
{(1,-1,1) (3,1,5) (1,1,2)}
It's always good to start with the definitions. A set of vectors, [tex]\{v_1, v_2, v_3, \cdot\cdot\cdot, v_n\}[/itex] are "independent" if and only if the only way we can have
$\displaystyle a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n= 0$ is to have $\displaystyle a_1= a_2= \cdot\cdot\cdot= a_n= 0$.
In this example, you would have
$\displaystyle a_1(1, -1, 1)+ a_2(3, 1, 5)+ a_3(1, 1, 2)$$\displaystyle = (a_1+ 3a_2+ a_3, -a_1+ a_2+ a_3, a_1+ 5a_2+ 2a_3)= (0, 0, 0)$.
That is, you have the three equations:
$\displaystyle a_1+ 3a_2+ a_3= 0$
$\displaystyle -a_1+ a_2+ a_3= 0$
$\displaystyle a_1+ 5a_2+ 2a_3= 0$
If the only solution to those three equations is $\displaystyle a_1= a_2=a_3= 0$ then the vectors are independent. If there are other solutions (a system of n homogeneous linear equations in n unknowns has a single unique solution, or has an infinite number of solutions) the vectors are dependent.