Lecture_3 - Lecture Note 3 Dr Jeff Chak-Fu WONG Department...

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Lecture Note 3 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1
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L INEAR E QUATIONS AND M ATRICES 1. Linear Systems 2. Matrices 3. Dot Product and Matrix Multiplication 4. Properties of Matrix Operations 5. Solutions of Linear Systems of Equations 6. The Inverse of A Matrix 7. LU-Factorization L INEAR E QUATIONS AND M ATRICES 2
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S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS 3
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Our ultimate goal is to study: 1. Solving linear system (a) Gauss-Jordan reduction - reduced row echelon method + elementary row operations (b) Gaussian elimination - row echelon method + elementary row operations + back substitution (c) Consistent/Inconsisent system i. If there is at least one solution, it is called consistent . ii. A system of equations that has no solutions is said to be inconsistent . 2. Homogeneous system (a) trivial solution (b) nontrivial solution S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS 4
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DEFINITION - An m × n matrix A is said to be in reduced row echelon form if it satisfies the following properties: (a) All zero rows, if there are any, appear at the bottom of the matrix. (b) The first nonzero entry from the left of a nonzero row is a 1 . This entry is called a leading one of its row. (c) For each nonzero row, the leading one appears to the right and below any leading one’s in preceding rows. (d) If a column contains a leading one, then all other entries in that column are zero. Note that a matrix in reduced row echelon form appears as a staircase (“echelon") pattern of leading ones descending from the upper left corner of the matrix. An m × n matrix satisfying properties (a) , (b) and (c) is said to be in row echelon form .
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A row echelon matrix takes its name from the French word “echelon" meaning “step". When a matrix is in row echelon form, the path formed by leading nonzero entries resembles a staircase. S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS 6
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Example 1 The following are matrices in reduced row echelon form since they satisfy properties (a), (b), (c) and (d): A = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , B = 1 0 0 0 - 2 4 0 1 0 0 4 8 0 0 0 1 7 - 2 0 0 0 0 0 0 0 0 0 0 0 0 and C = 1 2 0 0 1 0 0 1 2 3 0 0 0 0 0 . S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS 7
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The following matrices are not in reduced row echelon form (Why not?) D = 1 2 0 4 0 0 0 0 0 0 1 - 3 , E = 1 0 3 4 0 2 - 2 5 0 0 1 2 , F = 1 0 3 4 0 1 - 2 5 0 1 2 2 0 0 0 0 , G = 1 2 3 4 0 1 - 2 5 0 0 1 2 0 0 0 0 . S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS 8
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Example 2 The following are matrices in row echelon form: H = 1 5 0 2 - 2 4 0 1 0 3 4 8 0 0 0 1 7 - 2 0 0 0 0 0 0 0 0 0 0 0 0 , 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 and J = 0 0 1 3 5 7 9 0 0 0 0 1
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