FINAL EXAM ( min) MATH 65 SPRING 5 Student Name Student ID Instructor Section number (-digits, as in the table below) Mark Test and Section Number on your scantron! There are 5 questions on this exam. Each question is worth 8 points. Exam Rules. Students may not open the exam until instructed to do so.. Students must obey the orders and requests by all proctors, TAs, and lecturers.. No student may leave in the first min or in the last min of the exam. 4. Books, notes, calculators, or any electronic devices are not allowed on the exam, and they should not even be in sight in the exam room. Students may not look at anybody else s test, and may not communicate with anybody else except, if they have a question, with their TA or lecturer. 5. After time is called, the students have to put down all writing instruments and remain in their seats, while the TAs will collect the scantrons and the exams. 6. Any violation of these rules and any act of academic dishonesty may result in severe penalties. Additionally, all violators will be reported to the Office of the Dean of Students. I have read and understand the exam rules stated above: STUDENT SIGNATURE: Section Numbers: Bauman, Patricia: 7 Dadarlat, Marius: Gabrielov, Andrei: Ulrich, Bernd: Zhilan Julie Feng: 6 Noparstak, Jakob: 7 Ma, Linquan: (:am) and 7 (9:am) Li, Dan (:pm) and 5 (:pm) Kelleher, Daniel: (4:pm) and 4 (:pm) Gnang, Edinah (9:am) and 5 (:am) Xu, Xiang (MWF :pm) and 6 (MWF 4:pm) Zhang, Xu: 7 (TR :pm) and 7 (TR :pm)
() If we solve the equation: [ a + b c a c + a for a, b and c, the values of a and b are: = [ 4 A. a =, b = B. a = 7, b = 4 C. a = 5, b = D. a =, b = 4 E. There is no solution () Let A = 5 4 8. Compute the (,) entry of AT A. A. 7 B. 4 C. D. 4 E. 4
() Which of the following sets of vectors span R? A. B. C. D. E. (4) Which of the following is a basis for the subspace of R spanned by S =? A. B. C. D. E.
(5) If L : R R is a linear transformation such that ([ ) ([ ) L =, L = then a + b + c is equal to: and ([ L ) = a b c, A. 8 B. C. 7 D. 4 E. 5 (6) If y = ax + b is the least square fit line for the points (, ), (, ), (, 4), find a + b. A. B. 7/ C. D. 5 E. 9/ 4
(7) Consider the matrix t t t A = t t, t where t is a real number. Then A is nonsingular if and only if A. t B. t and t C. t and t D. t, t, and t E. t is any real number (8) For a nonsingular 6 6 matrix A, the determinant of the adjoint matrix adj(a) is A. det(a) B. det(a 5 ) C. det(a 4 ) D. det(a ) E. det(a ) 5
(9) Let V be the set of all strictly positive numbers in R, and let and be defined by a b = ab, for any a, b in V (that is, a, b are strictly positive numbers), c a = a c, for any a in V and c in R. Which of the following statements are true? (i) For any a, b in V and any c in R, c (a b) belongs to V. (ii) Under the given operations, the element is the real number. (iii) There is at least one element a in V for which there is no element a such that ( a) a =. A. (i) only B. (ii) only C. (i) and (ii) only D. (i) and (iii) only E. All of them () Let S = {v, v, v }, where v = v = v =. Which of the following statements are true? (i) A basis for span S is {v, v }. (ii) The vector u = 4 belongs to span S. (iii) S is a linearly dependent set. A. (i) only B. (iii) only C. (i) and (iii) only D. (ii) and (iii) only E. All of them 6
() Let W be the vector space spanned by the vectors: u = [, u = [, u = [, u 4 = [. Apply the Gram-Schmidt process to the vectors u, u, u, u 4 (in this order) to find an orthonormal basis w, w, w, w 4 of W. What is w? A. [ B. [ C. [ / / D. [ / / E. [ / () Consider the subspace W of R 4 : W = span {[, [, [, [ 4 }. What is the dimension of W? A. B. C. D. E. 4 7
() Consider the homogeneous linear system a + b + c + 4d + 5e = a + c + e = b + c + d = Then the dimension of the solution space is A. B. C. D. E. 4 (4) Suppose A is a 5 matrix such that rank(a) =. Which of the following is TRUE? A. The rank of A T is 5 B. The nullity of A T is C. Ax = only has trivial solution D. The rows of A are linearly dependent E. The columns of A are linearly dependent 8
(5) The eigenvectors of [ are [ and If x (t), x (t) satisfy x () =, x () = and [ x (t) x = (t) compute x () + x (). [ with eigenvalues and respectively. [ [ x (t) x (t) A. 4e B. 4e e C. e e D. e e E. e 4e (6) The dimension of the vector space of all 4 4 symmetric matrices with real entries is equal to: A. 6 B. 8 C. 9 D. E. 9
(7) What is the characteristic polynomial of A = A. (λ )(λ + ) B. (λ )(λ )(λ + ) C. (λ )(λ + )(λ ) D. (λ )(λ + )(λ + ) E. (λ + )(λ + )(λ ) (8) Let A = P DP where P = Then A equals ( ) (, D = ) and P = ( ). ( ) 9 56 A. 8 55 ( ) 9 56 B. 8 57 ( ) 9 55 C. 8 54 ( ) 4 56 D. 5 57 ( ) 4 56 E. 5 55
(9) Find the eigenvalues and associated eigenvectors of the following matrix: [ A = 5 [ A. λ =, λ = 4, x = 5 [ B λ =, λ = 4, x = 5 [ C. λ =, λ = 4, x = 5 [ D. λ =, λ = 4, x = 5 [ E. λ =, λ = 4, x = 5 [, x =, x =, x =, x =, x = [ [ [ [ () Which of the following matrices is not diagonalizable? [ A. [ B. [ C. [ D. [ E.
() For which values of a does Gaussian elimination applied to a A = a a 4 a a a fail to give three pivots (leading s)? A.,, B.,, C.,, D.,, 4 E.,, 4 () If then b + c must be equal to: a b c A = has inverse A = a b c, a b c A. /4 B. 5/4 C. /4 D. /4 E. 7/4
[ + i () Let A =. Then A i is given by [ + i A. i [ i B. + i [ ( + i)/ / C. / ( i)/ [ i D. + i E. A does not exist. (4) Let u and v be orthogonal vectors in R 5 such that u = 7, v =. Then u v equals A. 9 B. C. 5 D. 7 E. 6
(5) Let A be a 7 matrix such that its null space is spanned by the vectors, and. The rank of A is: A. B. C. D. 4 E. 6 4