A group of vectors are always linear dependent if la + mb = c if my memory is correct (it's been a while)
(there is an exception to this I think, maybe you need to ensure that they're not parallel - someone else can confirm this, but your textbook should tell you anyway)
l and m are just random constants btw
so yeah you will get:
4li + lj + 3k + 2mi - mj + 3mk = -4i + 2j + 6k
Then you equate all the vectors in the direction i, j and k
so for the ones with direction i:
4l + 2m = -4
2l + m = -4
j:
l - m = 2
k:
3l + 3m = 6
l + m = 2
So there we have 3 equations
Try and solve for l and m by choosing two of the equations
Say you picked the first two equations, once you get the solutions for l and m, sub them into the third equation to see if it actually equals 6
from observation, it looks like we won't find solutions that will satisfy all 3 of the equations simultaneously, so therefore, they are independent. If we did find solutions that satisfied all 3 of them, then the set of vectors would be linearly dependent
as you can tell, this is quite rushed, so I may have made some silly little errors or a massive one somewhere