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July 14th, 2015, 12:14 PM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124  Reduced Row Echelon Form is Unique Proof
No column in RREF containing a 1 and the rest zeroes can be changed by Elementary Row Operations without violating RREF.

July 17th, 2015, 09:54 AM  #2  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124  Quote:
Reduced Row Echelon Form, Elementary Row Operation Any matrix in RREF will have rows like: (1,x,0,0,x,x,x), (0,0,1,0,x,x,x), (0,0,0,1,x,x,x), (0,0,0,0,0,0,0), by construction. No ERO's can change (1,x,0,0), (0,0,1,0), (0,0,0,1) to a different RREF, for example (1,x,0,0) to (0,1,0,0). Any attempt to change the other x’s destroys the RREF. Therefore, RREF is unique under ERO’s. Last edited by zylo; July 17th, 2015 at 10:12 AM.  
October 16th, 2015, 09:03 AM  #3 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
Let R1 and R2 be reduced row echelon forms of A. Then R1<>R2 by elementary row operations. Therefore: R1 and R2 have same number of nonzero rows, same location of the (0,..0,1,0,..0)columns, and same elements in the nonzero rows. Examples $\displaystyle \begin{vmatrix} 1 &0 & 2 &1 \\ 0 & 1&3 &5 \\ 0 &0 & 0 & 0 \end{vmatrix} and\begin{vmatrix} 1 &4 &0 &0 \\ 0& 0 &1 &0 \\ 0 &0 &0 &1 \end{vmatrix}$ 

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