Given an $m\times n$ (with $n>m)$ matrix $M$ over a polynomial ring $R=k[x_1,...,x_n]$, suppose that every column of $M$ is an $R$-linear combination of $m$ specified columns. I would like to explicitly find these linear combinations. Are there any programs that would allow me to do this?
Of course, I can do this manually (and I have so far), but I am dealing with matrices with over a 100 polynomials and it's taking up a lot of time.