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2107test3sol_09 - MATH 2107A TEST 3 NOVEMBER 6 2009 This...

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MATH 2107A TEST 3 NOVEMBER 6, 2009 1 of 4 This test has two parts with a total of 30 marks. Part I has multiple choice questions, and Part II has long answer questions. The test cannot be taken out from the examination room. Only nonprogrammable calculators are allowed. Duration: 50 minutes. NAME (in ink): STUDENT NO (in ink): PART I: [6] Multiple choice questions. Circle the correct answer in ink. [2] 1. Let = = = 1 1 0 0 , 0 0 1 1 , 4 2 3 1 2 1 u u ν and let } , { 2 1 u u span W = . If = d c b a proj W ν , What is the value of d ? a) 1 b) 2 c) 3 d) 4 e) 5 [2] 2. Let A be a 3 3 × matrix such that . 3 0 1 , 1 1 1 ) ( = span A col If = 1 b a ν is in )) ( ( A col , What is the value of b a + ? a) -1 b) -3 c) 0 d) 1 e) 3 3. Let = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 A . You are given that 0 = λ is an eigenvalue of A . What is the dimension of the corresponding eigenspace 0 E ? a) 1 b) 2 c) 3 d) 4 e) 5 PART II: [24] Long answer questions. Show all your work. [8] 1. Let = 2 2 0 0 3 1 0 0 1 1 3 0 0 1 0 3 A . You are given that the eigenvalues of A are 3 , 1 = λ and 4. Determine whether A is diagonalizable and if so, find an invertible matrix P and a diagonal matrix D such that 1 = PDP A . I. Eigenvectors for 1 = λ = 0 0 0 0 3 2 0 0 1 1 4 0 0 1 0 4 ~ 3 2 0 0 3 2 0 0 1 1 4 0 0 1 0 4 ) 1 ( I A 4 x is a free variable.
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