lecture04 - Lecture 4 1.4 Gaussian Elimination 1.5...

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Chapter 1 Linear Systems & Gaussian Elimination 1 Lecture 4 1.4 Gaussian Elimination 1.5 Homogeneous Linear System
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Lecture 4 Announcement 2 Announcement ± Practice Session (answers and scores) ± Tutorial/lab balloting closed. Updated tutorial/lab timeslots available in IVLE workbin Contact VT if your tutorial/lab groups are not settled ± Tutorial class begins next week (week 3) Download your tutorial sets from IVLE Your tutor’s contact in IVLE ± Virtual Class (last call)
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Chapter 1 Linear Systems & Gaussian Elimination 3 From Last Time (1) Inconsistent system Consistent system Augmented column is a pivot column Augmented column is not a pivot column 0 0 0 " " % 0 0 0 0 " " % number of pivot columns = number of leading entries = number of non-zero rows number of variables greater than or equal to number of non-zero rows number of variables may be smaller than number of non-zero rows
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Chapter 1 Linear Systems & Gaussian Elimination 4 From Last Time (2) 3 2 1 6 6 6 5 5 5 4 3 2 4 3 2 1 = = = + + + + + + + + + + x x x x x x x x x x x x x non-pivot columns you can assign parameter to either one of x 5 or x 6 you can assign parameters to two among x 2 , x 3 , x 4
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Chapter 1 Linear Systems & Gaussian Elimination 5 Section 1.4 Gaussian Elimination Objective • How to use GE / GJE to solve indirect LS problems?
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Chapter 1 Linear Systems & Gaussian Elimination 6 Notation 1.4.8 When doing elementary row operations, we adopt the following notation: 1. The symbol cR i means multiply the i th row by the constant c ”. 2. The symbol R i ¨ R j means interchange the i th and the j th rows ”. 3. The symbol R i + cR j means add c times of the j th row to the i th row ”. i th row will be replaced don’t write: cR j + R i How to denote ERO?
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Chapter 1 Linear Systems & Gaussian Elimination 7 Example 1.4.9.1 What is the condition that must be satisfied by a , b , c so that the system of linear equations has at least one solution? = + = + = + c z y x b z y x a z y x 7 2 11 6 2 3 2 Look at its row echelon form Linear system with “unknown” constant terms
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Chapter 1 Linear Systems & Gaussian Elimination 8 Example 1.4.9.1 c b a 7 2 1 11 6 2 3 2 1 ⎯→ 1 2 2 R R c a b a 7 2 1 2 5 2 0 3 2 1 1 3 R R a c a b a 10 4 0 2 5 2 0 3 2 1 + 2 3 2 R R + a c b a b a 5 2 0 0 0 2 5 2 0 3 2 1 If 2 b + c 5 a 0, system has no solution It has (infinitely many) solutions if and only if 2 b + c 5 a = 0 .
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lecture04 - Lecture 4 1.4 Gaussian Elimination 1.5...

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