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Unformatted text preview: Be sure that this examination has 11 pages including this cover A The University of British Columbia
Final Examinations  April, 2003 Mathematics 152
All Sections Closed book examination  I Time; 2_5 hours Name Signature Student Number____._____ Instructor’s Name " Section Number Special Instructions: No aids allowed. Rules governing examinations 1 Each candidate should be prepared to produce his library/AMS
card upon request. 3. Read and observe the following rules:
No candidate shall be permitted to enter the examination room after the expi
ration of one half hour, or to leave during the ﬁrst half hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in
cases of supposed errors or ambiguities in examination questions.
CAUTION — Candidates guilty of any of the following or similar practices
Shall be immediately dismissed from the examination and shall be liable to
diBCiPIinary action. (3) Making use of any books, papers or memoranda, other than those au
th°rized by the examiners. 0’) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The
plea of accident or forgetfulness shall not be received. 3. Smoking is not permitted during examinations. ‘ Marks l
'11
i [20] April, 2003 1. MATH 152 Name Page 2 of .11 pages Let 3’ = [1, 2, 3]_and ‘17 = [3, 2, —1]. (a) Is the smallest positive angle between W and "15’ larger than a right angle or
smaller than a right angle? How do you know this? (b) Find a nonzero vector, 3:", that is perpendicular to both 72" and 717. (e) If E) is also perpendicular to both V and 717, how Would E’ be related to ’17
found in part (b) above? Continued on page 3 V April, 2003 MATH 152 Name M Page 3 of 11 pages (d) Find a plane through the point (—1, 0, 1) and containing a vector parallel to '17
and a vector parallel to ‘13. ' V .THE UNIVERSITY OF BRITISH COLUMBlA Counselling Services
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Vancouver, B.C. V6T 121 (é) Find thepoint P on the line 5' = 7&7 + t? nearest the point (—1,0, 1). Continued‘ on page 4 MATH 152 Name .____.______..____._____ Page 4 of _11 pages .prﬂ,‘ 2003 [10]. ,2]. Consider the system of equations 2m1—5m2~a:3=6
.'1:1 +3$2+4$3 =7
3:111 — 7m + 22:3 = 19. Write down the augmented matrix of this system. Convert it to a rowechelon form
and mark the pivot columns clearly. Then write down the general solution of the System. Continued on page 5 April, 2003 MATH 152 Name ____________ Page 5 of .11 pages Let
1 2 2
B = 2 3 O
——1 1 k where k is a real constant. (a) For what values of k is B invertible? ‘ (b) Find 31 "if k = 11. Continued on page 6 Apﬁl, 2003 MATH 152 Name ______+__ Page 6 of .11 pages , Let A be a 3 X 3 matrix and 71, .172 vectors in R3. It is given that the equation
[10] 4 _ _ —_, I u + a n n
A? = b 1 IS CODSISteIlt and A? = b 2 IS 1ncons1stent. (a) What is the largest number of pivots A can have? (b) Show that the equation A? = 2.53 — 71 is inconsistent. .4 Continued on page 7 April, 2003 MATH 152 Name _____—_—_——_ Page 7 of .11 pages [15] 5. Let C be the standard matrix of the twostep linear transformation of R3 that ﬁrst
projects all vectors into the plane with equation :1: = 0 and then rotates all vectors by
60° counter clockwise about the zaxis when viewed from above. (a) Find 0. (b) Find C‘f if it exists. If it does not exist, explain why. Continued on page 8 April, 2003 MATH 152 Name ________.____ Page 8 of _11 pages (c) Find one eigenvector and corresponding eigenvalue for C. Continued on page 9 April, 2003’ MATH 152 N ame __________——___ Page 9 of .11 pages [15] 6. Find the variables m(t) and y(t) if they satisfy the conditions that $’(t) = $(t)  W)
3/0?) = ﬂit) + W) with$=4andy=0whent=0. Continued on page 10 mil, 2003 MATH 152 Name .._._________ Page 10 of _11 pages Which of the following assertions is TRUE or FALSE. Justify your answers with a.
clear reason or a counterexample. ‘ (a) The value of det A is not changed by elementary row operation on A. i (b) An n x n matrix with real entries has n real eigenvalues (counting multiplicities). (c) A 3 x 3 matrix with eigenvalues 1, 5, 2 is diagonalizable. Continued on page 11 April, 2003 MATH 152 Name .______._._._._________ Page 11Lof11 pages (d) A 3 x 3 matrix with eigenvalues 1, 0, 1 is invertible. (e) If the 3 x 3 matrix A has eigenvalues —2, —1, 0, then all solutions of
. I 7
71? (t) = approach 0 as t —> oo. ...
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 Math, Linear Systems

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