Linear independance of (binary) vectors here is what i want to do.
I have 4 vectors (in $\mathbb{Z}^{12}_{2}$), lets say
$v_1 = (1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0)$
$v_2 = (1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0)$
$v_3 = (1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0)$
$v_4 = (1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1)$
These vectors are supposed to build a 4-dimensional vector subspace of $\mathbb{Z}^{12}_{2}$.
For that to be the case they have to be linearly independent. 
Now i want to check if they are. I could do that with a system of linear equations but accordung to my script i should: "[...] compute all 16 linear combinations in $\mathbb{Z}_{2}$ and check if any of them occur twice. If so, $v_1,v_2,v_3,v_4$" are not linearly independent."
I thought about that for over an hour now and i dont really get what i am supposed to do and whyit is correct. Would really appreciate some help.

@Thèophile 's answer is the best, I think. There's no way zeros can produce a $1$. So, $v_4$ is independent of the other vectors because of the last component.
Then $v_3$ is independent of $v_1$ and $v_2$ because of the sixth component. And finally, $v_2$ is independent of $v_1$ because of the third component.
