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

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

Question
Equations
asked 2021-01-02
Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions?

Answers (1)

2021-01-03
Step 1
Given:
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 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.
0

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