Suppose A and B are two mxn matrices such that rank (A) > rank (B) , then I have to prove that rowspace of B is a subset of rowspace of A.
What I tried was using property of 'Simplified Span Method' , i.e. Finding reduced row echelon form after using vectors as rows of matrix, that the final rref has linearly independent non zero rows. So A will have more linearly independent vectors than B as rank(A) > rank(B). But I do not know how to proceed if these vectors are different i.e.
Suppose n is 6, so maximum possible cardinal number for a subset of mxn matrices is 6 and A contains 3 of these and B contains 2 and the other one is left out. In this case, rowspace of B will not be a subset of rowspace of A.