Solving a system of linear equations using back substitution

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SUMMARY

This discussion focuses on solving systems of linear equations using back substitution and clarifying the concept of a target vector. A target vector represents the desired outcome in the context of linear equations, which can be used in both exact solutions and least-squares error scenarios. The conversation highlights that while back substitution typically involves a single target vector, it can also apply to multiple target vectors when seeking a least-squared-error solution. The mention of least-squares error emphasizes its significance in error minimization and quadratic minimization algorithms.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically systems of linear equations
  • Familiarity with back substitution techniques for solving linear equations
  • Knowledge of target vectors in the context of linear algebra
  • Basic understanding of least-squares error minimization methods
NEXT STEPS
  • Study the process of back substitution in detail with various examples
  • Learn about target vectors and their role in linear algebra
  • Research least-squares error minimization techniques and algorithms
  • Explore quadratic minimization methods and their applications in solving linear systems
USEFUL FOR

Students and professionals in mathematics, data science, and engineering who are working with linear algebra, particularly those solving systems of equations and optimizing solutions using back substitution and least-squares methods.

Jen2114
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Hello,
its been a while since I have taken linear algebra and I am having trouble understanding what a target vector is. I need to solve a system of linear equations in matrix form using back substitution and with different target vectors. I don't have a problem with back substitution, but I don't recall exactly what a target vector is and how I would use it with back substitution to solve the system. Could someone help me clarify this concept? Thanks!
 
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The only use of the term "target vector" that I have seen is where there is no feasible solution of the linear equations and a least squared error solution is desired.

But I imagine that there is no reason not to use that term also when there is an exact solution with no errors. In that case, a single solution with a single "target" vector would be the back-substitution process that you are familiar with.

Another possibility is that you are being asked to find a single vector of inputs that gives the best least-squared-error solution to multiple target vectors at the same time.

PS. The fact that I mention lease-squared-error is simply because that is the most common error minimization problem and a lot of effort has been exerted to find quadratic minimization algorithms to solve that problem. Other error metrics can also be used.
 

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