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July 14th, 2015, 11:14 AM   #1
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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.
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July 17th, 2015, 08:54 AM   #2
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Quote:
Originally Posted by zylo View Post
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.
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October 16th, 2015, 08:03 AM   #3
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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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