Understanding a Proof: Uniqueness of R Determined by W Explained

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In summary, the conversation discusses a question about the uniqueness of a reduced row echelon matrix R, given a subspace W and its corresponding basis. The author is struggling to understand the argument and is seeking clarification. The explanation suggests that the uniqueness of R is due to the linear independence of the basis vectors, which define a linear subspace. The reduced echelon form represents the equations for this linear function and is therefore unique.
  • #1
marcin w
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I've scanned a page out of my textbook and highlighted the portion of the proof I don't quite follow. I've been staring at this on and off for a day and for some reason it just doesn't click why the argument shows that R is uniquely determined by W. I see the author is proving an implication and it's converse, but I can't tie it together. I'd appreciate it if anyone could break it down for me a little more. Thanks.
 

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  • #2
The question, if anyone fancies answering it is this:

Given a subspace W, and some basis of W then why is the corresponding reduced row echelon matrix R unique.
 
  • #3
matt grime said:
The question, if anyone fancies answering it is this:

Given a subspace W, and some basis of W then why is the corresponding reduced row echelon matrix R unique.

row reduced matrix of what? the matrix of the basis?
 
  • #4
What else? You could have looked at the attachment, as well.
 
  • #5
because the column(basis) vectors are linearly independent
 
  • #6
given a system of equations, they define a linear subspace which can be viewed as the graph of a linear function on the unique subspace of coordinates furthest possible to the right, which is an isomorphic projection of the solution subspace.

the non trivial entries of the reduced echelon form are the equations for the function defined by this graph, hence are unique.

i don't expect everyone to understand this but some will.
 
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1. What is the purpose of a proof?

A proof is a logical demonstration that shows why a certain statement or theorem is true. It provides evidence and reasoning to support a claim or argument.

2. How do you approach understanding a proof?

To understand a proof, it is important to first read and familiarize yourself with the definitions, assumptions, and concepts used. Then, carefully follow the logical steps and reasoning presented in the proof.

3. What are some common types of proofs?

Some common types of proofs include direct proof, proof by contradiction, proof by induction, and proof by counterexample. Each type has its own structure and approach, but they all aim to logically demonstrate the truth of a statement.

4. How can I check if a proof is valid?

A valid proof should follow a clear and logical sequence of steps, use valid definitions and assumptions, and avoid fallacies or errors in reasoning. It is also helpful to double-check the proof with other sources or consult with other experts in the field.

5. What if I still don't understand a proof?

If you are having trouble understanding a proof, try breaking it down into smaller parts and following each step carefully. You can also try finding additional resources or seeking help from a teacher or fellow scientist. It is important to keep in mind that understanding a proof may take time and effort, but persisting through the challenge can lead to a deeper understanding of the concept.

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