Let $v_1,\dots,v_k\in \mathbb{Z}^n$. Is there a nice criterion for the existence of $v_{k+1},\ldots,v_n \in\mathbb{Z}^n$ such that $v_1,\ldots,v_n$ form a basis of $\mathbb{Z}^n$?
For $k=1$, if $v_1 = (a_1,\ldots,a_n)$, it is not hard to see that it is iff $\gcd(a_1,\ldots,a_n)=1$. For bigger $k$ I can find out the answer for specific $v_1,\ldots,v_k$ algorithmically, but I would like to know a general criterion if there is one.