Solving Line Space Problem with Matrices A & B

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Discussion Overview

The discussion revolves around determining the value of x in matrices A and B such that they have the same row space. Participants explore the definitions and implications of "line space," which leads to some confusion and clarification about the terminology used.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant presents matrices A and B and expresses confusion about the term "line space" and how to find x.
  • Another participant questions the definition of "line space," suggesting it may refer to a Liouville space, which may not be relevant to the problem.
  • A participant proposes that "row space" might be the intended meaning and suggests asking the instructor for clarification.
  • One participant confirms that "row space" is indeed the correct interpretation and expresses uncertainty about how to find the value of x.
  • Another participant advises to row reduce the matrices into echelon form and states that the resulting vectors must have the same span, indicating that finding the span relates to solving simultaneous equations.

Areas of Agreement / Disagreement

Participants generally agree that "line space" is likely a miscommunication and that "row space" is the intended term. However, there is no consensus on how to proceed with finding the value of x.

Contextual Notes

There is uncertainty regarding the definitions and implications of "line space" versus "row space," and the discussion includes references to specific mathematical techniques that may not be fully understood by all participants.

Cyannaca
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Consider the matrices

A=
(1 5 -3)
(2 x -3)

B=
(1 3 -1)
(3 x -1)

Find x so A and B have the same line space.
I tried to do this problem but I was stuck with

A=
(1 (x-5) 0)
(0 (x-10) 3)

B=
(1 3 -1)
(0 (x-9) 2)

Also, I'm not really sure of what they mean by same line space.
 
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If you don't know what the definition of "line space" is from your notes, then we're not guaranteed to know. Indeed the only definition I could find was that it was a Liouville space, which is a product of Hilbert spaces, so I don't think that's what you want.
 
Is it possible they mean "row" space? Or "column" space (unlikely, since they have the same column space in [itex]\mathbb{R}^2[/itex] regardless of the value of x)? Ask your intructor.
 
Yes, they mean row space. Sorry, I translated it from French a bit too fast I guess. Actually, I have to find the value of x so they have the same row space but I don't know how.
 
Row reduce into ecehlon form, the resulting vectors must have the same spane. Finding the span is the same as finding solutions to simultaneous equations. If you use reduced row echelon form the answer is even easier to read off. These topics will almost certainly bi in your notes, though I don't know the French terms for them (though I can guess what echelon is).
 

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