# Gausses Method

## Theorem
If a linear system is changed to another by one of these operations:

1. an equation is swapped with another
2. an equation has both sides multiplied by a non zero constant.
3. an equation is replaced by the sum of itself and a multiple of another

→ Then the two systems have the same set of solutions.

## Definition
The three operations from the theorem are the *elementary reduction operations*, or *row operations*, or *Gaussian operations*.

They are *swapping*, *multiplying by a scalar* and *row combination*

## Solution Space

### Systems without a unique solution

When using gausses method and in the echelon form the linear system produces something like $0 = -1$ we know the system does not have a solution.

### Systems with infinitely many solutions

When we do not have a single $variable = constant$ stage in the echelon form we know we have infinitely many solutions.

When one of the rows is $0 = 0$ or $1 = 1$ etc. we know that one of the input rows contained redundant information.