mat188f_2002_exam - University of Toronto FACULTY OF...

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Unformatted text preview: University of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATIONS, DECEMBER 2002 First Year - Programs 1,2,3,4,6,7,8,9 MAT 188H1F Linear Algebra SURNAME GIVEN NAME STUDENT NO. SIGNATURE INSTRUCTIONS: Non-programmable calculators permitted. Answer all questions. Present your solutions in the space provided; use the back of the preceding page if more space is required. TOTAL MARKS: 100 The value for each question is shown in parentheses after the question number. Page 1 of 10 Examiners D. Burbulla H. Bursztyn A. Kricker F. Recio MARKER’S REPORT 1. [30 marks: 5 marks for each part] Find the following: 1 2 4 (a) the inverse of 0 1 —1 1 0 2 (b) det COP-dbl I—ICDOI—l I—IMr—dw rP-HJOCA) Page 2 of 10 GOOD cam-c: CO“ V (c) the eigenvalues of the matrix ( (d) the coordinate vector of p(m) : —1 + 4x + 5:2 with respect to the basis B: {1+I,1+I2,$+I2} Of P2. Page 3 of 10 2 Cl'’ abuse 024:4— l (e) the values of a for which the matrix ( a ) is not invertible. Cl (f) the point on the plane with equation 5': + y + z = 2 closest to the point (3, 2, —1). Page 4 of 10 ‘2. [12 marks] Let W be the subspace of R4 consisting of all vectors of the form (a+c,b+c,a+2b+c,—a—b). Find an orthonormal basis of W. (Use the usual dot product in R4.) Page 5 of 10 3. [12 marks] Let W be the set of 3 x 3 matrices, A, satisfying the condition AT = —A. (a) [6 marks] Show that W is a. subspace of M33. (b) [6 marks] Find a basis for W and its dimension. Page 6 of 10 4. [12 marks] Let A = ( matnx D such that Page 7 of 10 5. [14 marks] Suppose A is a 3 x 3 invertible matrix with eigenvalues, 3, 1, and -—1. Find the following: (a) [4 marks] the eigenvalues of A“. (b) [5 marks] the eigenvalues of AT. (c) [5 marks] the eigenvalues of Adj(A). Page 8 of 10 6. [10 marks; 2 marks for each part] Suppose u and v are two non-zero vectors in R3. What does each of the following conditions imply about the linear independence or dependence of the set {11, v}? (a) u = 3v (b)au+bv=0=>a=b=0 (c)u'v=0 (d)uxv=0 (e) {u,v,u X v} spans R3 Page 9 of 10 I‘ 7. [10 marks: 5 marks for each part] Let B :: {w1,w2, . . . ,wn} be an orthonormal basis of an inner product Space V. Prove the following: (a) For any vectors u and v in V, (“a V) = x ' y: where x and y are the coordinate vectors of u and v, respectively, with respect to the basis B. (b) If x- is the coordinate vector of wi with respect to the basis B, then the set {x1,xz, . i . ,xn} is an orthonormal basis of R“, with respect to the usual dot product in R“. Page 10 of 10 ...
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mat188f_2002_exam - University of Toronto FACULTY OF...

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