What is the error in this proof of uniquness of row echelon form?

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Discussion Overview

The discussion centers on the proof of the uniqueness of row echelon form for matrices, specifically examining a proposed proof and identifying potential errors or gaps in reasoning. Participants explore the implications of solution sets for homogeneous equations and the relationship between different echelon forms.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant suggests that the proof relies on the uniqueness of the solution set for the homogeneous equation AX=0, arguing that if there are multiple row echelon forms, it implies multiple solution sets, which is a contradiction.
  • Another participant counters that while the solution set is non-empty, it does not imply uniqueness, indicating a flaw in the original proof's reasoning.
  • A different participant elaborates on the idea that two different echelon forms would lead to different solution sets, providing an example to illustrate how varying parameters in echelon forms affect the solutions.
  • One participant expresses confusion about the lack of detailed proofs in linear algebra texts, questioning why the proposed proof is not more widely accepted despite its simplicity.
  • Another participant encourages more detailed writing of the proof to enhance understanding, suggesting that clarity and thoroughness are essential for grasping the concept.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the proposed proof and its clarity. While some appreciate the idea behind the proof, others highlight the need for more rigorous detail and challenge the assumptions made regarding uniqueness.

Contextual Notes

Participants note that the proof lacks sufficient detail and does not adequately address the nuances of solution sets in relation to different echelon forms. There is also an indication that the discussion may benefit from a deeper exploration of linear transformations and vector space concepts.

Who May Find This Useful

This discussion may be useful for students and educators in linear algebra, particularly those interested in the properties of row echelon forms and the implications of solution sets in linear systems.

mveritas
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Let's prove the uniqueness of row echelon form (Suppose we already knew existence)

First, for any m*n matrix A, think about homogeneous equation AX=0.

Obviously, AX=0 has a solution X=0, so its solution set is not empty.

And A's row echelon form has same solution set. So if there are more than 2 row echelon

forms, it's contradiction because it means AX=0 has more than 2 solution set.

I don't know where's the error in this proof...
 
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mveritas said:
Let's prove the uniqueness of row echelon form (Suppose we already knew existence)

First, for any m*n matrix A, think about homogeneous equation AX=0.

Obviously, AX=0 has a solution X=0, so its solution set is not empty and unique.
The obvious solution means that its solutions set is non-empty. It says nothing about uniqueness.

And A's row echelon form has same solution set. So if there are more than 2 row echelon

forms, it's contradiction because it means AX=0 has more than 2 solution set.




I don't know where's the error in this proof...
 
NoNO, I mean 'solution set' is unique...

hmm i'll change my paragraph...
 
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this seems like a nice idea for a proof. I.e. any two echelon forms would have the same solution set as the original system, but two different echelon forms "obviously" have different solution sets.

e.g, suppose the echelon form of a 3x3 system were of rank one, and reduced to one non zero row of form [1 3 4 ]. Then the solutions have y,z arbitrary and x = -3y -4z. Any other such echelon form would look like [ 1 a b] and have solutions with y,z, arbitrary and x = -ay - bz. These could not be the same unless a = 3, and b = 4.

Stated geometrically, the set of solutions defines the graph of a mapping from the space of free variables, to the space of pivot variables. And the non pivot columns of the echelon form are the values of this mapping at the standard basis vectors (0,...0,1,0,...,0) of the space of free variables.

I.e. in any solution vector, the values of the free variables determine the values of the pivot variables, and the echelon form reveals exactly how they do so.

Indeed this argument makes it so obvious that the echelon form is unique that it is a mystery that it is not explained in every book. In fact this makes it seem that understanding why it is unique, is equivalent to just understanding what the echelon system of equations says.
 
//To mathwonk

So, you mean this is the correct proof? (inspite lack of explain...)

I wondered why so many linear algebra books don't prove it in this way... (most of them prove it by using linear transformation or concept of vector space...)

So I thought it maybe wrong proof... it's so simple!

Thank you!
 
i said the idea was good. but it should be written and worked out in considerably more detail. the more detail you can give, the better you yourself will understand it. try writing it up in 2-3 pages. and convince one of your friends.
 

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