Fix a $d$-dimensional discrete addition subgroup $L\subset \mathbb{R}^n$. Given $v_1,\dots, v_k\in L$ whose integer combinations span $L$, how can one find a group basis $b_1,\dots,b_d\in L$?

I suppose this problem is equivalent to designing an $A\in \mathbb{R}^{n\times d}$ where the following equation has an integer-matrix solution for $U$: $$AU=\begin{bmatrix}v_1,\dots, v_k\end{bmatrix}$$.

I think the following restricted version is easier: say $k=d+1$ and $c_1v_1+\dots + c_k v_k = 0$ for some $c_1,\dots,c_k\in \mathbb{Z}$.