A 4 by 5 matrix, with 4 rows and 5 columns maps a vector in to . The "range" of such a matrix, the subspace of into which all of is mapped, cannot have dimension larger than 4. If, in addition, one row is all "0"s (or the matrix can be row reduced to that), then the range cannot have dimension greater than 3. If, after row-reduction, that is the only row of all "0"s, the range has dimension 3. Of course, that would mean that the nullspace has dimension 5- 3= 2.