aflacatn

2020-10-21

Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions?

Explain.

Explain.

Adnaan Franks

Skilled2020-10-22Added 92 answers

An overdetermined homogeneous system of linear equations.

Step 2

The homogeneous system of linear equations is said to be overdetermined if the number of equations m is more than the number of unknowns n.

If all the equations are linearly independent and m >n then there is no solution.

Step 3

If the system of equation is consistent, then, in this case, there is either one solution or set of solutions.

If all the equations are not linearly independent and s out of m equations are linearly independent, and if

(i) s>n, then there is no solution.

(ii) If s=n, then the system has either one solution or no solution.

(iii) If s<n, then the system has infinitely many solutions.

For example:

Consider the homogeneous system of linear equations.

x+y=0, 2x+3y=0, 3x+2y=0

Here the number of equations is m=3 and the number of unknowns n=2.

As m>n, therefore, the system is overdetermined.

x=y=0 satisfy the equations.

Step 2

The homogeneous system of linear equations is said to be overdetermined if the number of equations m is more than the number of unknowns n.

If all the equations are linearly independent and m >n then there is no solution.

Step 3

If the system of equation is consistent, then, in this case, there is either one solution or set of solutions.

If all the equations are not linearly independent and s out of m equations are linearly independent, and if

(i) s>n, then there is no solution.

(ii) If s=n, then the system has either one solution or no solution.

(iii) If s<n, then the system has infinitely many solutions.

For example:

Consider the homogeneous system of linear equations.

x+y=0, 2x+3y=0, 3x+2y=0

Here the number of equations is m=3 and the number of unknowns n=2.

As m>n, therefore, the system is overdetermined.

x=y=0 satisfy the equations.

Jeffrey Jordon

Expert2021-10-15Added 2605 answers

Step 1If the system is overdetermined there are more equations than variables. Let's assume that there are m equations and n variables and that m>n(1) If there are more linearly independent equations than there are variables, then the system is inconsistent.(2) If there are exactly n linearly independent equations, then the system has either no solution or a unique solution.(3) If there are less linearly independent equations than there are variables, then the system can be reduced to an underdetermined system which either doesn't have any or has infinitely many solutions.

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