If I start with a polynomial ideal generated by a set of polynomials $f_i$, which is not a standard basis for the ideal, and then I obtain a standard basis for the ideal to be a set of polynomials $g_j$, how can I figure out how to write each $g_j$ as a linear combination of $f_i$?
Because $g_j$ is a standard basis, without much difficulty we can write each $f_i$ as a linear combination of $f_i = \sum_j M_{ij} g_j$, where each $M_{ij}$ is a polynomial. So we have a "matrix of polynomials" describing the "change of presentation" from $g_j$ to $f_i$ (not sure the correct phrasing, as $f_i$ is not a basis, so this is not a change of "basis"... maybe change of generators?).
With this matrix $M_{ij}$ in hand, is there some way I can "invert" it to get the matrix $N_{ij}$, such that $g_j = \sum_i N_{ji} f_i$?
If I knew the proper phrases to describe this process, I could probably search better. So that would be helpful as well.
This question is very related:
Use / take advantage of Gröbner basis $ G = (g_j)$ to write $f \in I = \langle f_i \rangle$ as a $k$-linear combination of the polynomials in $I$?
It shows how one could, in principle, calculate this by carrying extra information along while calculating the standard basis. But I am not calculating the standard basis by hand. The example I am working with is large, and while computer packages can calculate the basis fairly easily (in about 10 minutes with a few GB of memory), they fail if I try to calculate the matrix simultaneously: it quickly runs through all the memory on my computer trying to carry around extra information in case it is needed for the matrix. That is why I want to know if there is some way to use just the basis itself after it is calculated (and the $M_{ij}$ which is not hard to generate from it).
Writing $g$ as a linear combination (with polynomial coefficients) of $f_1, \ldots,f_r$ is sometimes called "lifting", e.g. lift in Singular and lift in SageMath (which uses Singular). The resulting coefficients can be called witnesses to (or a certificate of) the membership of $g$ in the ideal generated by the $f_i$.
I'm not sure exactly how the computation goes, but I think it amounts to finding syzygies of $f_1,\ldots,f_r,g$ where the coefficient of $g$ is a unit (or more generally syzygies of $f_1,\ldots,f_r,g_1,\ldots,g_s$ where the coefficients of $g_j$ are units). The internal implementation in Singular is found in idLift, which someone might be able to decipher.