I hope that the following problem isn't actually elementary (at least, for the sake of the fact that I'm posting it here), and I apologize if it is. I did try hard to solve it first.

Let $V$ be a $\mathbb{Q}$-vector subspace of $\mathbb{Q}^n$, and let $G = V \cap \mathbb{Z}^n$. Does there exist a linearly independent generating set for $G$ (i.e. a subset of $G$ such that every element of $G$ can be expressed uniquely as as $\mathbb{Z}$-linear combination of elements of this subset)? Is there an algorithm to find it (given a basis for $V$)?