27 lines
1.0 KiB
Markdown
27 lines
1.0 KiB
Markdown
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# Gausses Method
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## Theorem
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If a linear system is changed to another by one of these operations:
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1. an equation is swapped with another
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2. an equation has both sides multiplied by a non zero constant.
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3. an equation is replaced by the sum of itself and a multiple of another
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→ Then the two systems have the same set of solutions.
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## Definition
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The three operations from the theorem are the *elementary reduction operations*, or *row operations*, or *Gaussian operations*.
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They are *swapping*, *multiplying by a scalar* and *row combination*
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## Solution Space
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### Systems without a unique solution
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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.
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### Systems with infinitely many solutions
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When we do not have a single $variable = constant$ stage in the echelon form we know we have infinitely many solutions.
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When one of the rows is $0 = 0$ or $1 = 1$ etc. we know that one of the input rows contained redundant information.
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