midterm3-practice-sol - Math 415 Midterm 3 Thursday Circle...

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Math 415 - Midterm 3 Thursday, November 20, 2014 Circle your section: Philipp Hieronymi 2pm 3pm Armin Straub 9am 11am Name: NetID: UIN: Problem 0. [1 point] Write down the number of your discussion section (for instance, AD2 or ADH) and the first name of your TA (Allen, Anton, Mahmood, Michael, Nathan, Pouyan, Tigran, Travis). Section: TA: To be completed by the grader: 0 1 2 3 4 5 6 Shorts /1 /? /? /? /? /? /? /? /? Good luck! 1
Instructions No notes, personal aids or calculators are permitted. This exam consists of ? pages. Take a moment to make sure you have all pages. You have 75 minutes. Answer all questions in the space provided. If you require more space to write your answer, you may continue on the back of the page (make it clear if you do). Explain your work! Little or no points will be given for a correct answer with no explanation of how you got it. In particular, you have to write down all row operations for full credit. Problem 1. Let A = 1 - 1 1 0 1 1 1 2 and b = 5 0 5 10 . Find a least squares solution of A x = b . Solution. We have to solve A T A ˆ x = A T b : A T A = 1 1 1 1 - 1 0 1 2 1 - 1 1 0 1 1 1 2 = 4 2 2 6 and, A T b = 1 1 1 1 - 1 0 1 2 5 0 5 10 = 20 20 . Since 4 2 20 2 6 20 R 2 R 2 - 1 / 2 R 1 ---------→ 4 2 20 0 5 10 , we obtain ˆ x = 4 2 . Problem 2. Let W = span 0 1 0 1 , 0 1 1 1 . (a) Find an orthonormal basis for W . (b) What is the orthogonal projection of 1 2 1 0 onto W ? (c) Write 1 0 0 0 as the sum of a vector in W and a vector in W . (d) Find the projection matrix corresponding to orthogonal projection onto W . 2
Solution. (a) We apply Gram-Schmidt to { v 1 , v 2 } = 0 1 0 1 , 0 1 1 1 . We have: u 1 = v 1 k v 1 k = 0 1 0 1 k 0 1 0 1 k = 0 1 2 0 1 2 and, u 2 = v 2 - ( u 1 · v 2 ) u 1 k v 2 - ( u 1 · v 2 ) u 1 k = 0 1 1 1 - ( 0 1 2 0 1 2 · 0 1 1 1 ) 0 1 2 0 1 2 k 0 1 1 1 - ( 0 1 2 0 1 2 · 0 1 1 1 ) 0 1 2 0 1 2 k = 0 0 1 0 k 0 0 1 0 k = 0 0 1 0 Hence, 0 1 2 0 1 2 , 0 0 1 0 is an orthonormal basis for W .

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