1 Can a system of 20 linear algebraic equations with 14 unkn

1) Can a system of 20 linear algebraic equations with 14 unknowns have a 2-parameter family of solutions?

2) Can a system of 20 linear algebraic equations with 14 unknowns have a 14-parameter family of solutions?

3) Can a system of 20 linear algebraic equations with 14 unknowns have a 16-parameter family of solutions?

Solution

We know that:

Given a system of m equations in n unknowns:

             (Thus if there is a unique solution we must have m n.)

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.

In view of the above:

1) Can a system of 20 linear algebraic equations with 14 unknowns have a 2-parameter family of solutions? No as 20 > 14.

2) Can a system of 20 linear algebraic equations with 14 unknowns have a 14-parameter family of solutions? No, as 20 > 14.

3) Can a system of 20 linear algebraic equations with 14 unknowns have a 16-parameter family of solutions? No, as 20 > 14.

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.