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

[mveritas]

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...

 

[HallsofIvy]

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...

 

[mveritas]

NoNO, I mean 'solution set' is unique...

hmm i'll change my paragraph...

 

[mathwonk]

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.

 

[mveritas]

//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!

 

[mathwonk]

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.