Trying to understand least squares estimates

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Discussion Overview

The discussion centers around understanding the mathematical processes involved in least squares estimates, particularly in relation to solving systems of equations represented as Ax=b, where b is not in the column space of A. The scope includes mathematical reasoning and conceptual clarification.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant seeks clarification on the mathematical steps needed to arrive at a specific solution related to least squares estimates.
  • Another participant suggests that the inquiry may be focused on matrix multiplication, expressing confusion about the original question.
  • A third participant explains that least squares provides "best possible solutions" for systems where the vector b is not in the column space of matrix A, emphasizing the minimization of the L^2 distance between Ax and b, which corresponds to the orthogonal projection of b onto Ax.
  • A participant acknowledges the clarification regarding matrix multiplication and indicates a willingness to review it.

Areas of Agreement / Disagreement

The discussion includes some agreement on the importance of matrix multiplication in the context of least squares, but there is no consensus on the specific question being asked or the steps required to arrive at the solution.

Contextual Notes

There may be limitations in the clarity of the original question, as well as potential dependencies on definitions related to least squares and matrix operations.

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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    Screenshot at Feb 26 00-11-44.png
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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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