... for both and c to yield, c cThere is one zero row in the left-hand side, and the rank of and that of is , the number of nonzero rows. The residual system is (compatible), and , so the system ... This is the point of the following theorem.Theorem 2.4.4 The number of linearly independent columns of any matrix is equal to the number of itsindependent rows, and where null .Proof. We have ... implementednumerically.Let be the index of the last nonzero row of . Since this is the number of independent rows of , is the rank of . It is also the rank of , because and admit exactly the same solutions and are...