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.
Best Answer
You can immediately see that $v_4$ is independent from the others since its 12th element is $1$, whereas the others have $0$ in that position; there is no way that a combination of $0$s can create a $1$.
Similarly, $v_2$ and $v_3$ are independent of the others based on the 3rd and 2nd elements respectively.
You can then see that $v_1$ is not a combination of the others. Therefore they are all independent, and no need to compute any of the 16 combinations.