Understanding a Proof: Uniqueness of R Determined by W Explained

[marcin w]

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.

 

[matt grime]

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.

 

[ice109]

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?

 

[matt grime]

What else? You could have looked at the attachment, as well.

 

[ice109]

because the column(basis) vectors are linearly independent

 

[mathwonk]

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.