Linear spanning sets for matrix spaces

Necessary and sufficient conditions are given on matrices A, B and S  , having entries in some field FF and suitable dimensions, such that the linear span of the terms AiSBjAiSBj over FF is equal to the whole matrix space.This result is then used to determine the cardinality of subsets of F[A]SF[B]F[A]SF[B] when FF is a finite field.