I'm an academic mathematician.  While trying to verify a counterexample to something in my research, the following computational problem has arisen:Fix a vector space $V = \mathbb{Q}^n$.  Here $n$ is something on the order of $20,000$.  I have a procedure that spits out a sequence of vectors $\vec{v}_1,\vec{v}_2,\ldots$ in $V$.  The vectors $\vec{v}_i$ all have integer entries; these entries are bounded in size (by something on the order of 20 or 30).  For each $k \geq 2$, I need to decide if $\vec{v}_k$ is in the $\mathbb{Q}$-span of the set $\{\vec{v}_1,\ldots,\vec{v}_{k-1}\}$.What algorithm should I use for this?  The naive thing to do would be to just use Gauss-Jordan elimination (or, if I want to mostly stick with integers, compute the Hermite normal form).  However, experiments I have done show that the size of the numbers that arise when you do this explode in complexity even for much smaller problems.  This accords with stuff I have found while searching the literature, though I don't know the computer science literature very well and have trouble determining the state of the art in stuff like this.
Efficient algorithms for dealing with linear algebra over the rationals
algorithms;linear algebra