Unique solution of an overdetermined system

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In summary, to determine the number of solutions for a consistent linear system with more equations than unknowns, you need to find the number of independent equations. This is done by comparing the number of variables to the number of independent equations, with the number of free variables being the difference between the two. This can be calculated by writing the coefficient matrix and row-reducing it, with the number of non-zero rows representing the number of independent equations.
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Shaybay92
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If I want to know how many solutions a consistent linear system with more equations than unknowns has, how do I tell? Obviously there is either 1 solution of infinite solutions. Can you have a free variable in this case? I'm confused how to find out whether a system will give a unique solution.
 
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You need to determine how many independent equations there are. If the system really is consistent, then there must be no more independent equations than unknown variables. That is, if n is the number of variables and m is the number of independent equations, then [itex]n\ge m[/itex]. The number of free variables is n- m.

If you write the coefficient matrix for the system and row-reduce, the number of independent equations is the number of non-zero rows.
 

1. What is an overdetermined system?

An overdetermined system is a set of equations or constraints that has more equations than unknown variables. This means that there are more conditions to be satisfied than there are variables to be determined.

2. Can an overdetermined system have a unique solution?

Yes, it is possible for an overdetermined system to have a unique solution. This can happen when the extra equations are not independent and can be derived from the other equations in the system. In this case, the system is considered to be consistent and has a unique solution.

3. How is the unique solution of an overdetermined system calculated?

The unique solution of an overdetermined system is typically calculated using methods such as least squares or matrix inversion. These methods involve finding the best possible solution that satisfies the given equations, even if it does not satisfy all of them perfectly.

4. Are there any real-world applications of overdetermined systems?

Yes, overdetermined systems have many real-world applications, particularly in fields such as engineering, physics, and economics. They are often used to model complex systems with more constraints than unknown variables, such as in optimization problems or data analysis.

5. What happens if an overdetermined system does not have a unique solution?

If an overdetermined system does not have a unique solution, it is considered to be inconsistent. This means that there is no set of values that can satisfy all of the equations in the system. In this case, the system is either under-constrained or contradictory, and it is not possible to find a unique solution.

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