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 July 14th, 2015, 11:14 AM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 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, 08:54 AM   #2
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Quote:
 Originally Posted by zylo No column in RREF containing a 1 and the rest zeroes can be changed by Elementary Row Operations without violating RREF.
An explanation is in order:

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 09:12 AM.

 October 16th, 2015, 08:03 AM #3 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 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 non-zero rows, same location of the (0,..0,1,0,..0)columns, and same elements in the non-zero 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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