Linear spanning sets for matrix spaces
  Necessary and sufficient conditions are given on matrices $A$, $B$ and $S$, having entries in some field $\mathbb F$ and suitable dimensions, such that the linear span of the terms $A^iSB^j$ over $\mathbb F$ is equal to the whole matrix space. This result is then used to determine the cardinality of subsets of $\mathbb F[A]S\mathbb F[B]$ when $\mathbb F$ is a finite field. 
Giacomo Micheli, Joachim Rosenthal, Paolo Vettori
math.RA