◀ ▲ ▶Branches / Algebra / Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE
The following lemma ensures that the elementary Gaussian operations indeed do not change the solutions of an SLE:
Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE
Let an SLE be given in the form of an extended coefficient matrix $A\gamma$ and let $S$ be the set of its solutions. Then $S$ is an invariant of the elementary Gaussian operations.
In other words, if we apply any finite series of these operations $o_1,o_2,\ldots,o_k$ to $A\gamma$, leading to the new SLEs
 SLE$_1:=A^{(1)}\gamma^{(1)}:=o_1(A\gamma)$ after the first opertion $o_1,$
 SLE$_2:=A^{(2)}\gamma^{(2)}:=o_2(A^{(1)}\gamma^{(1)})$ after the second opertion $o_2,$
 SLE$_3:=A^{(3)}\gamma^{(3)}:=o_3(A^{2)}\gamma^{(2)})$ after the third opertion $o_3,$
 ...
 SLE$_k:=A^{(k)}\gamma^{(k)}:=o_k(A^{(k1)}\gamma^{(k1)})$ after the $k$th opertion $o_k,$
then the solutions $S$ of SLE$_k$ (if any), will be exactly the same as the solutions as the original SLE.
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References
Bibliography
 Knabner, P; Barth, W.: "Lineare Algebra  Grundlagen und Anwendungen", Springer Spektrum, 2013