Chap3 - CHAPTER 3 System of Linear Equations 3.1 Gauss...

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CHAPTER 3 System of Linear Equations 3.1 Gauss Elimination Method 3.2 Algorithm of Gauss Elimination 3.3 Crout’s Method 3.4 Inverse of Matrix 3.5 Condition Numbers and Errors 3.6 Iterative Methods 1
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Notations • The above system of equations can be written in matrix form as Ax=b . A is called the coefficient matrix .(of order n) • b is called the price vector /(constant vector) • [A|b] is called the augmented matrix . 2
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Consistency (Solvability) • The linear system of equations Ax=b has a solution, or said to be consistent if and only if Rank{A}=Rank{A|b} • A system is inconsistent when Rank{A}<Rank{A|b} Rank{A} is the maximum number of linearly independent rows of A. Rank can be found by using ERO (Elementary Row Operations) ERO # of rows with at least one nonzero entry 3
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Elementary row operations • The following operations applied to the augmented matrix [A|b], yield an equivalent linear system. Interchanges: The order of two rows can be changed. Scaling: Multiplying a row by a nonzero constant. Replacement: The row can be replaced by the sum of that row and a nonzero multiple of any other row. 4
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An inconsistent example = 5 4 4 2 2 1 2 1 x x ERO:Multiply the first row with -2 and add to the second row 0 0 2 1 Rank{A}=1 Then, this system of equations is not solvable 3 4 0 2 0 1 Rank{A|b}=2 5
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Uniqueness of solutions • The system has a unique solution if and only if Rank{A}=Rank{A|b}=n, n is the order of the system . • Such systems are called full-rank systems . 6
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Full-rank systems • If Rank{A}=n Det{A} 0 A is nonsingular so invertible Unique solution . = 2 4 1 1 2 1 2 1 x x 7
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Rank deficient matrices • If Rank{A}=m<n Det{A} = 0 A is singular so not invertible, infinite number of solutions ( n-m free variables) under-determined system. = 8 4 4 2 2 1 2 1 x x Consistent so solvable Rank{A}=Rank{A|b}=1 8
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