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Gaussian Elimination and Back Substitution

Jim Lambers MAT 610. Summer Session 2009-10. Lecture 4 Notes These notes correspond to Sections and in the text. Gaussian Elimination and Back Substitution The basic idea behind methods for solving a system of linear equations is to reduce them to linear equations involving a single unknown, because such equations are trivial to solve. Such a reduction is achieved by manipulating the equations in the system in such a way that the solution does not change, but unknowns are eliminated from selected equations until, nally, we obtain an equation involving only a single unknown. These manipulations are called elementary row operations, and they are de ned as follows: Multiplying both sides of an equation by a scalar Reordering the equations by interchanging both sides of the th and th equation in the system Replacing equation by the sum of equation and a multiple of both sides of equation.

The process of eliminating variables from the equations, or, equivalently, zeroing entries of the corresponding matrix, in order to reduce the system to upper …

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