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--- elementary_row_operation [2015/01/27 12:18] – created rupert
+++ elementary_row_operation [2015/03/26 10:12] (current) – rupert
@@ Line 1: / Line 1: @@
-If we perform one of the following operations on a system of linear equations:
-  - list the equations in a different order; or
+Recall that when we form the [[augmented matrix]] of a linear system, each equation in the system becomes a row of the matrix. So we can translate the [[elementary operations on a linear system|elementary operations on the linear system]] into corresponding operations on the rows of the matrix. We get three different types:
-  - multiply one of the equations by a non-zero real number; or
-  - replace equation $j$ by equation $j$ ${}+{}$ $c\times {}$ equation $i$, where $c$ is a non-zero real number,
-then the new system will have exactly the same solutions as the original system. For operations of type 1, this is trivial, and for operations of types 2 and 3, this is because
+  - change the order of the rows of the matrix;
+  - multiply one of the rows of the matrix by a non-zero real number;
+  - replace row $j$ by "row $j$ ${}+{}$ $c\times {}$ (row $i$)", where $c$ is a non-zero real number and $i\ne j$.
-  * we are doing the same thing to the left hand side and the right hand side of each equation, so any solution to the original system will also be a solution to the new system; and
+The system of linear equations corresponding to these matrices will then have exactly the same solutions.
-  * these operations are reversible, using operations of the same type, so any solution to the new system will also be a solution to the original system.
-We call these operations of types 1, 2 or 3 **elementary operations** on the linear system.
+We call these operations **elementary row operations** or **EROs** on the matrix.