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Unformatted text preview: UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATIONS, APRIL 2001
MAT 188 S — LINEAR ALGEBRA. FIRST YEAR: TPROGRAM
EXAMINER: FELIX J. RECIO INSTRUCTIONS: QUESTION
NUMBER l. ATTEMPT ALL QUESTIONS.
2. SHOW AND EXPLAIN YOUR WORK IN ALL QUESTIONS. 3. GIVE YOUR ANSWERS IN THE SPACE PROVIDED.
USE BOTH SIDES OF PAPER, IF NECESSARY. 4. DO NOT TEAR OUT ANY PAGES. 5. USE OF NONPROGRAMMABLE POCKET CALCULATORS,
BUT NO OTHER AIDS ARE PERMITTED. 6. THIS EXAM CONSISTS OF EIGHT QUESTIONS. THE VALUE
OF EACH QUESTION IS INDICATED (IN BRACKETS) BY
THE QUESTION NUMBER. 7. THIS EXAM IS WORTH 50% OF YOUR FINAL GRADE.
8. TIME ALLOWED: 2 V2 HOURS. 9. PLEASE \VRITE YOUR NAME, YOUR STUDENT NUMBER,
AND YOUR SIGNATURE IN THE SPACE PROVIDED AT THE
BOTTOM OF TIIIS PAGE. NAME: . (FAMILY NAME. PLEASE PRINT.) (GIVEN NAME.) STUDENT No.2 SIGNATURE: I I II H I l M Page 2 1, Considerthe points A(2 ,0‘ l ), [3(2 , l ,0),and C( l . l ,k)‘ a) (5 marks) Find the values of k , if any, for which the line that passes through the points A and C is
parallel to the plane 2 x — 5 y + 3 : = l b) (5 marks) Find the values of k , if any, for which the line that passes through the points A and C
contains the point (4 ,  2 ,8). c) (5 marks) Find the values of k , if any, for which the plane that passes thrOugh the points A , B , and
C is perpendicular to the line with parametric equations 1: = — 1 +31, y = — 2 I , and : = 5 — 2 I. d) (5 marks) Find the values of k , il‘anv, for which the angle between the vectors :4? and TC is It! 6 . e) (5 marks) Find the values of k , if any, for which the volume of the parallelepiped generated by the
vectors (:4. ,55, and (7C is 7. f) (5 marks) Find the values of I: , if any, for which the distance from the point C to the line that passes through the points A and B is 2. { I H H Page 3 {I { D Page 4 r. + 2x3 + 14 = 3
. x + 3. + r = 1
2. (15 marks) Solve the lmcar system: 2 4 5
r, — .13 + 2;:1 — r5 = 2
.r, + x2 + 2x} — IS = — 2 I
V
3) (15 marks) Consider the linear system: ’
— I
2x + my Find the values of the constant a , ifany, for which:
a) The system has no solutions. b) The system has exactly one solution. c) The system has exactly two solutions. d) The system has inﬁnitely many solutions. tomb 'J U ll Page 5 U r’ I O —l
4.(15marks)Considcrthematrices.4=l l 2 0 and B: \011 Find all matrices M, ifany, for which AT— 2 M = [3 ~ .4 M. [J I“) to DJ LAL) _.l\.)l\) Page 6 { I 5. (15 marks) Let A = Compute det M . Page 7 and let M be another 4 x 4 matrix such that det ( A A43 ) = l . Page 8 6. Let S bethesubspaceof R'a generatedbythevectors v.=( I ,0,— 1 ,2), V2=(0, I ,0,]),
v;=(1,2,—l,0),v4=(—l,l,l,3),and vs=( I, 0,} ,0).
a) (10 marks) Determine the dimension ofthe subspace S and ﬁnd a basis for S .
b) (5 marks) 15 the vector v 5 a linear combination ofthe vectors v . , v 2 , v 3 , and v4 ? Why or why not? c) (5 marks) 15 the vector v =( l , — 3 , — 1 , 1 ) one ofthe vectors in S ? Why or why not? h Page 9 7. (20 marks) Let C [  ] , 1 ] denote the inner product space consisting of all real valued functions which
are continuous on the interval [ 1 , 1 ] ,with the inner product deﬁned as ( f, g ) = II] f(x)g(x)dx. Find an onhonormal basis for the subspace of C [  l , l ] spanned by the set { ] , 3 + x . 2 x + 3 x2 } . 1 I 8. (20 marks) Given the matrix A =
D such that P“AP=D. —2 Page 10 . Find an invertible matrix P and a diagonal matrix U Page 11 9. Determine, in each of the following cases, whether the given proposition is true or false. Give and brieﬂy
explain your reasons in each case. a) (5 marks) If A is any 3 x 3 matrix such that detA = 3 ,then det( Adj A )= 27. b) (Smarks) [f M is any 5 x 7 matrix such that the rank of M is 3 , then the dimension ofthe solution
spaceof A x=0 is 2‘ c) (5 marks) The set consisting of all polynomials p ( .r ) = a + h x + c x 2 such that a = b c is a subspace
of P2. cl) (5 marks) lf 2. is an eigenvalue ofthc square matrix A , then £2 is an eigenvalue of the matrix A3 . ...
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