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I've learned about elementary row operations, there is one of them that seems a little bit weird to me: The row switching. It seems that a system of equations:

$$\begin{eqnarray*} {a_1x+b_1y+c_1z}&=&{d_1} \\ {a_2x+b_2y+c_2z}&=&{d_2} \\ {a_3x+b_3y+c_3z}&=&{d_3} \end{eqnarray*}$$

Can be represented by an augmented matrix:

$$\begin{pmatrix} {a_1}&{b_1}&{c_1}&{d_1}\\ {a_2}&{b_2}&{c_2}&{d_2}\\ {a_3}&{b_3}&{c_3}&{d_3} \end{pmatrix}$$

And then switching rows would ammount to just shift the order of the equations and hence, wouldn't change the solutions. This is a little confusing, I've been thinking about it's utility and until now, it seems that this ERO is just a visual aid for computing the other ERO's on matrices. Is that it or switching rows is something deeper and I still didn't notice?

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For numerical algorithm, switching rows is called pivoting and improves stability compared to just "creating a $1$" in the row that happens to be in the current position. For manual / exact computations, there is absolutely no difference.

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  • $\begingroup$ Can you point me to some example in which this computation yields problems? It's not needed to type it if you don't want, referencing some book or article would be good. $\endgroup$ – Billy Rubina Aug 7 '15 at 3:01
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    $\begingroup$ @Voyska A quick google search brought this lecture up: math.okstate.edu/people/binegar/4513-F98/4513-l12.pdf I think it conveys the problem. $\endgroup$ – AlexR Aug 7 '15 at 7:36
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For a field $K$ (take $K = \mathbb{R}$ if you're confused by this), the group $GL_n(K)$ is generated by elementary matrices.

Elementary matrices are by definition the matrices obtained by applying a single elementary row operation to the identity matrix.

This is quite a nice statement which wouldn't be true if you ignored the row-switching operation.

In a way, you might say that the nontriviality of the row switching operation corresponds to the fact that the matrix associated to a linear transformation depends not only on a choice of basis for the vector space, but also on the order of the basis vectors. Permuting the basis vectors corresponds to a nontrivial linear transformation, and the group of such permutations is $S_n$, which of course is generated by transpositions (corresponding to row-switches).

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