MST10030 notes wiki

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If we perform one of the following operations on a system of linear equations:

1. list the equations in a different order; or
2. multiply one of the equations by a non-zero real number; or
3. 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

• 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
• 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.