Why is the minimum norm solution commonly used for underdetermined systems?

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In summary, an underdetermined system is a system of equations with fewer equations than variables, leading to multiple possible solutions. This is in contrast to an overdetermined system, which has more equations than variables and may have no solution or a unique solution. Some real-life examples of underdetermined systems include economic forecasting, data analysis, and signal processing. While these systems cannot be solved for a unique solution, techniques such as Gaussian elimination, least squares method, and linear programming can be used to find optimal solutions. However, there are limitations to solving underdetermined systems, such as the possibility of infinite solutions and the lack of constraints leading to potentially inaccurate solutions.
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ayvee
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Hi folks. This is something I've been wondering about for a while now. Is there a reason why taking the minimum norm solution is the standard thing to do for an underdetermined system, or is it just an arbitrary tie-breaking rule?
 
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Just an arbitrary rule.
 

What is an underdetermined system?

An underdetermined system is a system of equations where there are fewer equations than variables. This means that there are multiple solutions that satisfy the equations, making it impossible to determine a unique solution.

What is the difference between an underdetermined system and an overdetermined system?

An underdetermined system has fewer equations than variables, while an overdetermined system has more equations than variables. This means that an underdetermined system has multiple solutions, while an overdetermined system may have no solution or a unique solution.

What are some real-life examples of underdetermined systems?

Some examples of underdetermined systems include economic forecasting, data analysis, and signal processing. In these situations, there may be more variables than data points, leading to an underdetermined system.

How do you solve an underdetermined system?

An underdetermined system cannot be solved for a unique solution. However, it can be solved using various techniques such as Gaussian elimination, least squares method, or using linear programming techniques to find the optimal solution.

What are the limitations of solving underdetermined systems?

One of the main limitations is that there may be an infinite number of solutions, making it difficult to determine the best solution. Additionally, the solutions may not be unique or may not accurately represent the real-life situation due to the lack of constraints in the system.

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