# Prove linear dependence

1. Sep 30, 2004

### vsage

I've already found the answer to this solution but I want to check my methods because the class is very proof-based and the professor likes to take off points for style in proofs on tests:

5. Is {(1, 4, -6), (1, 5, 8), (2, 1, 1), (0, 1, 0)} a linearly independent subset of R^3? Justify your answer.

Obviously the answer is no because R^3 has a dimension of 3 and if you're given 4 generating vectors then one isn't necessary. However, I tried creating the linear system and got stuck trying to prove that for this system:

a + b + 2c = 0
4a + 5b + c + d = 0
-6a + 8b + c = 0

that at least one of a, b, c and d is nonzero and

a(1, 4, -6) + b(1, 5, 8) + c(2, 1, 1) + d(0, 1, 0) = 0

works for at least one nonzero a, b, c, d. Little help? The professor alluded to the fact that you can use Gaussian Reduction but he said we wouldn't learn how to do that for another chapter but he regularly uses it in class. The only way I know of it is studying ahead a year back or so but how do I reduce this?

Last edited by a moderator: Sep 30, 2004
2. Sep 30, 2004

### vsage

Nevermind I got it.. Proved that a(1, 4, -6) + b(1, 5, 8) + c(2, 1, 1) = (0, -1, 0)