Hello

While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$ I am would like to find $w_1, \ldots w_n \in \mathbb{Z}^m$ such that

$$1.) \ \ \ \ \ \ \ \mathbb{Z}v_1+\dots +\mathbb{Z}v_n \ \subseteq \mathbb{Z}w_1 + \dots + \mathbb{Z}w_n $$

$$2.) \ \ \ \ \ \ \ \ |w_i|^2 \ \text{is small/ small as possible/ a lot smaller that $|v_j|^2$}$$

$$ 3.) \ \ \ \ \ \ \text{The vectorspaces spanned by $\{v_i\}$ and $\{w_i\}$ are identical}$$

In other words I want that if each $v_i$ satisfies that $v_{i,1}\alpha_i + \dots + v_{i,m}\alpha_m=0$ for a fixed collection of $\alpha_i \in \mathbb{R}$ the same remains true for the $w_i$ (this is of course the real condition I want).

If I wanted the lattices spanned by $v_i$ and $w_i$ to be identical LLL would clearly be the natural approach, but since I am not require this, this seems to not make my $v_i$ nearly as small as I can achieve under these weaker conditions. Does there exists an algorithm, approach, idea which could come up with a such a basis $v_i$ given the $w_i$.

EDIT : I missed the last condition and had some trouble updating correctly. Hope it makes sense and is not completely trivial now.