Gaussian elimination
Gaussian elimination is an algorithm in linear algebra for determining the solutions of a system of linear equations. First we do a forward elimination: Gaussian elimination reduces a given system to either triangular. Next, we do a backward elimination to solve the linear system
Problem
We want to solve the following linear system of
In the matrix form, we have
with
Example of resolution
Consider the following system:
with
- First step of the Gaussian elimination: we eliminate
in the lines and :
- Second step of the Gaussian elimination: we eliminate
in the line :
- By backward elimination we solve the linear system, we get the solution
:
Gaussian Elimination algorithm: forward elimination and triangular form
Let
Gaussian Elimination algorithm: backward elimination
Now the matrix
with
We use a backward elimination for solving
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