(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.)
- How do you define the rank of a matrix whose entries are polynomials in $K[X]$?
- If you assign some element of $K$ in the entries of such a matrix, what is the rank of the produced matrix (in $M_{mn}(K)$)? Is it larger, equal, or smaller than that of the original matrix?
EDIT: Here $K$ denotes an arbitrary field, but mostly I'm interested in $\mathbb{R}$ and $\mathbb{F}_p$.