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Lecture Notes 3

# Lecture Notes 3 - Lecture 03 Lecture 03 Systems of Linear...

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Lecture 03 1 September 11, 2007 Outline: Gaussian Elimination ( Ax =b Ux =c ) 1) Review mechanics 2) Review modes of failures 3) The general pattern of GE 3) GE using Matrices (E and P) 4) General Rules of Matrix-Matrix Operations Addition: (A+B) Matrix Multiplication: AB Matrix Powers: A p Lecture 03: Systems of Linear Equations #2: Gaussian Elimination Gaussian Elimination on a 3x3 system of equations: Example ( with row exchange): x + 2y + 4z = 1 2x + 4y + 2z = 2 6x + 10y - z = 8 Matrix Vector form: Augmented Matrix form: Gaussian Elimination on a 3x3 system of equations: [ [ 1 2 4 1 2 4 2 2 6 10 -1 8 A b Step 1: Step 2: Step 3: Step 4: Example 2: Example 1: Failure of Gaussian Elimination (review): The overall Pattern of Gaussian Elimination: A recursive Algorithm that reduces Ax =b to Ux =c If successful: U will contain n pivots on the diagonal (unique solution) If Fails: U will contain at least 1 zero on the diagonal (no or solutions) Gaussian Elimination using matrices:

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Lecture Notes 3 - Lecture 03 Lecture 03 Systems of Linear...

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