A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).