On rank of a matrix whose entries are polynomials (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 $\mathbb{F}_p$.
 A: *

*You could use the Smith normal form, a generalization of Gaussian elimination for PID.

*You can expect the rank to be at most the rank of the original matrix.
A: I assume that you are taking $K$ to be an arbitrary field here? Generally, the best way to view an m by n matrix with entries in a ring $R$ is as a linear map from a finite-dimensional $R$-module to another. 
$R$-modules are nothing to be afraid of; they're like a normal vector space but your scalars don't have to be from a field, they can be from a general ring (like $K[X]$ in this case).
The rank is then the dimension of the image of this linear map as an $R$-module. Hence it is the smallest number of elements $x_1,...,x_k$ such that everything in the image of the linear map can be written as $a_1x_1+\cdots+a_kx_k$ where $a_i$ are in $K[X]$.
EDIT: To make things simpler, you may as well assume that the coefficients are in the field of rational functions $K(X)$ (where the elements are quotients of regular polynomials). Then the image genuinely is a vector space, and all is well.
