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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: Nonprogrammable 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 COPdbl
I—ICDOI—l
I—IMr—dw
rPHJOCA) Page 2 of 10 GOOD
camc: 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 nonzero 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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 Spring '07
 Burbula
 Linear Algebra

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