128ahw12sum10 - MATH 128A SUMMER 2010 HOMEWORK 12 SOLUTION...

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MATH 128A, SUMMER 2010, HOMEWORK 12 SOLUTION BENJAMIN JOHNSON Homework 12: Due Wednesday, August 4 6.1; 1a, 3b 6.2; 1a, 3a, 7a 6.3; 1b, 5c, 6c 6.4; 1b, 12a section 6.1 1. For each of the following linear systems, obtain a solution by graphical methods, if possible. Explain the result from a geometrical standpoint. a. x 2 + x 2 = 3 x 1 - x 2 = 0 Solution: Figure 1. Plot using Mathematica 0.5 1.0 1.5 2.0 x1 0.5 1.0 1.5 2.0 x2 From the graph, it looks like the solution is x 1 = 1, x 2 = 1. 3. Use Gaussian Elimination with backward substitution and two-digit rounding arithmetic to solve the following linear systems. Do not reorder the equations. (The exact solution to each system is x 1 = 1, x 2 = - 1, x 3 = 3.) b. 4 x 1 + x 2 + 2 x 3 = 9 2 x 1 + 4 x 2 - x 3 = - 5 x 1 + x 2 - 3 x 3 = - 9 Solution: For the Gaussian Elimination part: Date : August 4, 2010. 1
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2 BENJAMIN JOHNSON 4 1 2 | 9 2 4 - 1 | - 5 1 1 - 3 | - 9 -→ R 2 - 0 . 5 R 1 R 2 R 3 - 0 . 25 R 1 R 3 4 1 2 | 9 0 3 . 5 - 2 | - 9 . 5 0 0 . 75 - 3 . 5 | - 11 R 3 - 0 . 21 R 2 R 3 4 1 2 | 9 0 3 . 5 - 2 | - 9 . 5 0 0 - 3 . 1 | - 9 . 0 and for the backwards substitution part: x 3 = - 9 . 0 - 3 . 1 = 2 . 9 x 2 = - 9 . 5+2 · 2 . 9 3 . 5 = - 1 . 1 x 1 = 9 - 2 · (2 . 9) - 1 · ( - 1 . 1) 4 = 1 . 1 section 6.2 1. Find the row interchanges that are required to solve the following linear systems using Algorithm 6.1. a.
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128ahw12sum10 - MATH 128A SUMMER 2010 HOMEWORK 12 SOLUTION...

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