1.0 KiB
1.0 KiB
Gausses Method
Theorem
If a linear system is changed to another by one of these operations:
- an equation is swapped with another
- an equation has both sides multiplied by a non zero constant.
- 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.