I was reading the current issue of AMM when I came across this term "space of relations", which I don't understand. Basically, we are given 9 vectors (0,1,0,0), (2,0,0,0), (1,1,0,0), (3,0,0,0), (2,1,0,0), (1,0,0,1), (1,2,0,0), (2,0,1,0), (3,1,0,0). We consider these vectors in $\mathbb{F}_2$ and we want to determine in how many ways can select some of them so that the sum is (0,0,0,0).
In the example above, the rank of the 9 exponent vectors over $\mathbb{F}_2$ is 4. Hence the space of relations has dimension 9 − 4 = 5 and thus we can construct $2^5 − 1 = 31$ non-trivial relations in this way.
In group theory, I know that a relation is some combination of elements that yields the identity. So, I'm guessing that a relation in this context is some combination of vectors that sum up to 0. But why is it of dimension 9 - 4 = 5?