Question

For each system described below say as much as possible about each system’s solution
set. Note which theorems you are using to reach your conclusions.
(a) A consistent system of 8 equations in 5 variables.
(b) A consistent system of 5 equations in 8 variables.
(c) A system of 4 equations in 9 variables.
(d) A system with 15 equations in 35 variables.
(e) A system with 8 equations and 5 variables. The reduced row-echelon form of the
augmented matrix of the system has 6 pivot columns.

Answer 1

THEOREM 1:If there are more variables (n) than the number of equations (m), then the number of solutions of the system is either 0 or infinite. If a system is homogeneous, then it has the zero solution and thus a homogeneous system is always consistent.

THEOREM 2:If the number of unknowns are greater than number of equations then the columns of the coefficient matrix are linearly dependent.

THEOREM 3:For the system of equation to be consistent ,rank(augmented matrix)=rank(coefficient matrix)

Overdetermined system of equations: A system of equations is considered overdetermined if there are more equations than unknowns.An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients.However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others.

(A)Here,there are more equations (8) than the number of variables (5) and the system is also consistent,means system of equation is overdetermined (but consistent),it means atleast one equation occurs more than once.(USAGE OF Overdetermined system of equations ).

(B)Here,number of variables (8) are more than the number of equations (5),so the columns of coefficient matrix are linearly dependents.Also,system of equations is consistent and columns ar linearly dependent,it means system of equations have infinite solutions (USAGE OF Theorem 2)

(C)Here,number of variables (4) are more than the number of equations (9),so the columns of coefficient matrix are linearly dependents.(USAGE OF Theorem 2)

(D)Here,number of variables (15) are more than the number of equations (35),so the columns of coefficient matrix are linearly dependents.(USAGE OF Theorem 2)

(e)Here,there are more equations (8) than the number of variables (5) ,means system of equation is overdetermined.Since , reduced row-echelon form of the augmented matrix of the system has 6 pivot columns.So,rank(augmented matrix)=6,while the rank(coefficient matrix)=5.Hence,rank(augmented matrix)=6 is not equal to rank(coefficient matrix)=5 thus making the system of equation inconsistent.(USAGE OF THEOREM 3).