Rank of a subgroup of $\mathbb{Z}^3$ given a generating set $(2,-2,0)$, $(0,4,-4)$, and $(5,0,-5)$? Is there some standard approach to finding the rank of a subgroup given a generating set?
In particular, I'm considering the subgroup of $\mathbb{Z}^3$ generated by $v_1=(2,-2,0)$, $v_2=(0,4,-4)$, and $v_3=(5,0,-5)$. This is a submodule of a free $\mathbb{Z}$-module of rank $3$, so the subgroup has rank at most $3$. Also, it is clear that any two distinct generating vectors are independent. So the rank is at least $2$. However, $10v_1+5v_2-4v_3=0$, so the generating set is not a basis. 
Is there some procedure to determine what the rank is?

The submodule generated by $v_i$ is contained in the rank $2$ submodule of $\mathbb Z^3$ consisting of those tuples whose entries add up to zero.
