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CS210_lect04

# CS210_lect04 - Lecture 4 Gaussian Elimination The basic...

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Lecture 4 Gaussian Elimination The basic idea behind the Gaussian elimination method is to transform general linear system of equations b Ax = into triangular form. To do this we need to replace selected nonzero entries of matrix by zeros. This can be accomplished by taking linear combinations of rows. Let us first see how Gaussian elimination works. Consider the following system of equations 0 1 2 2 6 3 2 1 3 2 1 3 2 1 = + = + = + + x x x x x x x x x The matrix representation of this system is = = 0 1 6 1 1 1 1 2 2 1 1 1 3 2 1 x x x b Ax Now there are several operations that one can perform on a system of equations, without changing its solution. 1. We can replace any equation by a non-zero constant times the original equation 2. We can replace any equation by its sum with another equation 3. We can carry out the above operations any number of times. In the matrix language the operations we apply on augmented matrix [ ][ ] = 0 1 6 1 1 1 1 2 2 1 1 1 b A Thus for example we can add –2 times the first row to the second row and –1 times the first row to the third row to obtain the following equivalent system

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CS210_lect04 - Lecture 4 Gaussian Elimination The basic...

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