Linear Algebra October 2017 var1-4.pdf - Linear Algebra...

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Linear Algebra, Final Exam 2017 NAME: Q.# 1 2 3 4 5 6 7 8 9 10 ANSWER SECTION I: MULTIPLE CHOICE Directions: Each of the following problems is followed by five choices. Select the best choice and put the corresponding mark into the answer sheet above. Don’t forget to put in your name. Incorrect answers are penalized by 0 . 25 points, while correct answers are rewarded with +1 . 00 points. There is no penalty for answers left blank. You may write in any part of this booklet. There are additional pages you may use as a scrap paper. If you wish, you may detach them gently, without unstapling the booklet. This booklet is to be turned in at the end of the exam. Calculators may NOT be used at any part of the exam. Time reserved for this section is: 45 min. 1. Quadratic λ -quasipolynomials are functions of the form ( a 0 + a 1 x + a 2 x 2 ) e λx . Which of the following is true about / ∂x , the linear operator of differentiation by x , in the vector space of quadratic λ -quasipolynomials? I. / ∂x is diagonalizable II. / ∂x has an eigenvalue of algebraic multiplicity 3 III. / ∂x has an eigenvalue of geometric multiplicity 1 (A) I (B) I and II (C) I and III (D) II and III (E) I, II, and III 2. Let e 1 , . . . , e n be a set of vectors in a linear vector space V . The Gram-Schmidt orthogonalization method assures that there ex- ist vectors f 1 , . . . , f n such that I. f i and f j are orthogonal for i negationslash = j II. L ( e 1 , . . . , e n ) = L ( f 1 , . . . , f n ) III. L ( e 1 , . . . , e k ) = L ( f 1 , . . . , f k ) for all k = 1 . . .n (A) I (B) II (C) III (D) I and II (E) I and III 3. The length of the projection of the vector x = ( 1 , 2 , 6) onto the linear hull of vectors e 1 = ( 1 , 1 , 1) and e 2 = (0 , 1 , 1) is 4. Linear operator A is such that A ( e 1 ) = e 1 + e 2 , A ( e 2 ) = e 1 e 3 , and A ( e 3 ) = 2 e 2 + e 3 . Then A - 1 (2 e 1 e 2 ) =? 5. Let A be a non-degenerate n × n matrix. Which of the following MUST be true about A ? Proceed to the next page
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Linear Algebra, Final Exam 2017 I. All eigenvalues of A are non-zero II. Eigenvectors that correspond to the same eigenvalue are linearly dependent III. Eigenvectors that correspond to different eigenvalues are linearly independent 6. Let A be a matrix such that A n = 0 for some n > 0 (such a matrix is called nilpotent ). Which of the following MUST be true? I. λ = 0 is an eigenvalue of A II. A doesn’t have eigenvalues other than λ = 0 III. If A is diagonalizable, then A = 0 (A) I only (B) I and II (C) I and III (D) II and III (E) I, II, and III
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