Undergrad Trying to understand least squares estimates

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SUMMARY

The discussion centers on understanding least squares estimates, specifically in the context of solving the equation Ax = b, where vector b is not in the column space of matrix A. The least squares method provides the best possible solutions by minimizing the L² distance ||Ax - b||₂, which represents the orthogonal projection of b onto the column space of A. Participants emphasized the importance of matrix multiplication in this process, indicating it as a foundational step in deriving the least squares solution.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly matrix operations.
  • Familiarity with the least squares method and its applications.
  • Knowledge of vector spaces and orthogonal projections.
  • Basic proficiency in mathematical notation and terminology.
NEXT STEPS
  • Study matrix multiplication techniques and their implications in linear algebra.
  • Explore the derivation of least squares estimates in detail.
  • Learn about orthogonal projections and their significance in vector spaces.
  • Investigate applications of least squares in data fitting and regression analysis.
USEFUL FOR

Students and professionals in mathematics, data science, and engineering who are looking to deepen their understanding of least squares estimates and their applications in solving linear systems.

Nastya
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Hi, I'm trying to understand which mathematical actions I need to perform to be able to arrive at the solution shown in the uploaded picture. Thank you.
 

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As a general statement, least squares allows you to obtain " best possible solutions " to systems like :

Ax=b , where b is not in the column space of A. This statement leads to a system like the one you attached to your post. You find the values of x that minimize the
## L^2 ## distance ## ||Ax -b ||_2 ##, and this is the orthogonal projection of b onto Ax.
 
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mfb said:
Are you just looking for the matrix multiplication? If not, I don't understand what you are asking.
Yes, thank you. I will review the matrix multiplication.
 

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