Finding a basis for the integer lattice points in a subspace When writing this answer, one subgoal involved finding all integer solutions to the equations $\sum_{i=1}^5x_i=\sum_{i=1}^5ix_i=0$ in $\Bbb Z^5$. Since this is a linear system, the solutions form a lattice, and so should be spanned by a few integer vectors, and since it is a 5D space with two constraints, I expect to find $3$ such vectors. With a little guesswork, I came up with the solutions
$$v_1=(3,-4,0,0,1)\qquad v_2=(2,-3,0,1,0)\qquad v_3=(1,-2,1,0,0)$$
and I wish to conclude that all solutions are of the form $n_1v_1+n_2v_2+n_3v_3$ for $n_k\in\Bbb Z$. Now if this were a vector space this would be the end of it (using say $n_k\in\Bbb R$ instead), since you can easily verify that the vectors are independent, but as a $\Bbb Z$-module I'm not sure how to prove that all integer lattice points in the subspace are covered here. Is there a general technique for this type of problem (which is to say I want to span the set of $\Bbb Z^n$ solutions to some collection of linear equations)?
 A: Intuitively: Since your basis vectors are linearly independent, they form a basis. The only problem is if they were spanning only a sub-lattice of the lattice you are describing and not the whole thing. This would be the case if you had chosen for example:
$$v_1'=2v_1=(6,-8,0,0,2)\qquad v_2=(2,-3,0,1,0)\qquad v_3=(1,-2,1,0,0)$$
Since there are obviously no other lattice points between 0 and the 3 points you have selected, everything is fine and the selected basis of $v_1,v_2,v_3$ spans the whole lattice.
Formally: Let $x=(x_1,x_2,x_3,x_4,x_5)\in L$ where L is your lattice. We need to prove that x can be written in the form $x=av_1+bv_2+cv_3=(3a+2b+c,-4a-3b-2c,c,b,a)$. Obviously, $x_3=c,x_4=b,x_5=a$. We also know that
$$x_1+x_2=-x_3-x_4-x_5=-c-b-a$$
and
$$x_1+2x_2=-3x_3-4x_4-5x_5=-3c-4b-5a$$
from the defining properties of the lattice, which can be solved to give 
$$x_1=3a+2b+c \quad\mbox{and}\quad x_2=-4a-3b-2c$$
as above. This way we have proven that any lattice point is of this form. Since we also knew that any vector of this form is a lattice point (because it satisfies the defining relations), we are done.
