A is an m by n matrix of rank r Suppose there are right side

A is an m by n matrix of rank r. Suppose there are right sides b for which Ax. = b has no solution. What are all inequalities (

Solution

(a) We know that:

             Given a system of m equations in n unknowns:

·         If m < n then the number of parameters in the solution, if any, of the equation Ax = b, will be at least n – m.   

·         If m > n the system is called overprescribed.

Overprescribed systems either have no solution or they contain redundancy. Redundancy means that we can find (mn) equations which can be dropped without affecting the solution. If a system of equations has no solution it is called inconsistent. If a system of equations has at least one solution it is called consistent.

A system of a certain number of linear equations in a certain number of unknowns will have parametric solutions only if the number of unknowns exceeds the number of equations.

If the mxn matrix A has rank r, it means that the RREF of A has r non-zero rows. For m > n, there is a possibility that the system Ax = b has no solution. Thus, if the system Ax = b has no solution, we must have r m , m > n .

(b) If A is a m x n matrix, then the number of leading variables in the linear system equivalent to Ax = 0   is min (m, n). The number of non-zero equations in the echelon form of the system is equal to the number of leading entries. The number of free variables plus the number of leading variables = n, the number of columns of A. The homogenous system Ax = 0 has non-trivial solutions if and only if there are free variables. If A is a m x n matrix, then A^T is a n x m matrix. The linear system equivalent to A^T y = 0 will have n equations in m variables. Thus the equation A^T y = 0 will have non-trivial solutions if there are free variables i.e. n > m and also r > m, where r is the rank of A^T