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111-2007&amp;2008-1-M20-Dec2007

111-2007&amp;2008-1-M20-Dec2007 - KUWAIT UNIVERSITY...

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Unformatted text preview: KUWAIT UNIVERSITY LINEAR ALGEBRA MATH 111 Dept. of Math. & Comp. Sci. Second Exam December 12, 2007 Calculators and mobile phones are not allowed. Answer all of the following questions and explain all details. Total mark: 25 points. Duration: 90 minutes 1 3 —1 0 0 2 —2 1 1. (1+2+1 pts.) Let A _ _1 _3 2 2 . 1 1 1 1 (a) Calculate the cofactor A12. (b) Find detG—A). (c) If possible, ﬁnd det(adj(A‘1)), without calculating A“. 2. (2+2 pts.) Let the vectors X :— (1,2, —3) and Y = (5—13, %, i). (a) Find all vectors of length 2 that are parallel to X. (b) Show that for any vector Z E R3, the vectors Y and Z — (Z - Y)Y are orthogonal. 3. (2+2+2 pts.) Let P(0,0, 1), Q(1,0,0) and R(1, 1,0) be given points. (a) Show that P, Q, R are not on the same line. (b) Find the area of the triangle PQR. (c) Write an equation for the plane which contains P and Q, and it is parallel to the line ﬁg—1 2 y:+1_1 = g. 4. (2 pts.) Determine whether R2 with the operations (x,y)69(m’,y’)= (m+w’,y+y’), 69%;!) = (any) is a vector space, or not. 5. (1+2 pts.) (a) Check whether W = {(a, b, c, d) : d = a — 20+ 1} is a subspace of R4. (b) Let A be an m x n matrix. Show that the set of solutions of the system AX :: 0 is a subspace of IR". 6. (4 X 1.5 pts.) Answer each of the following as TRUE or FALSE. Justify your answer. (a) The permutation 52341 is odd. (b) If X and Y are any two parallel vectors in IR", then X - Y > 0. 7 b 2a + 3b 4b — a __ (”H d 2c+3d 4d~c “ (d) The line :3 : 3,3; = 1 +t,z \$=7t,y:O,z=O. = 2, then 22. a C = 2 + 3t is perpendicular to the line 09 €01 c/r/sz 1a A ~(1)H20¢z 4:9, 11" ‘ '12?) C ad A-4= -13-.L=—4- l: ,1, 4 In )1 [A ] 'IAP ‘4 ' 43-10 --"— ‘ 0 = - .. H_ o 3'21R+e,_,23-:1 3112—4 Soldihl=_11z}m=_g__ A--1‘321_1+R00 13. . 11111404271 12,1 :llw?‘ [LE/Z; (flielif7etfo-t faith/9‘ ﬁé/Y' TAM Ur—i‘X) we! He’m M=V§Xf&-=%t)'( Pui' VTZ|+l=zqmo€ claim. trig: , 7- ' . 7. a. mum, \/.(Z - (2-507) = y=z - (\/»Z))I\/n H = \/.z — \/~Z =0) becwe WW4- 71‘» a M z — (2-7) we MW“ at. ﬁ=0mmL anal PWFf/Llé e?“ 15% ##e ‘65"6 L032 ﬁzz/3?)“: 76:?) 3:012:4’ZL- 7-5944 R(41710)¢L‘f-Aecme 4920' ...7 —a \f-Z. 5- Willa—1). 30) [six 57%) =(41020-7'Mz/9M4m"): i-IIFQXFRIFE' e. Let U,[email protected],-f, B7) 0414/ EFF—4f? (77077—1) . 71m ﬁe mo—‘rrmaﬂ vex”— fa ﬁe ‘pﬂme in 77?: CENT]: = (4,5,4): and x+53 +9 «:0 f9 '1') e M. 4. (Miami (¢+A)e>(rly)=czug) mt @v®(1:3))®Cd @(m 5)) = (20:1) 69 (x73) scum 13):}:(125g, 73.9): mg. (x;3)=/=(0;0). mm CR:@,O) I}; W)?" a V8442 space. {a CI. W I‘o molL a owéorace #510 R4 Awe [0, a, 0,0) éW ‘5' Wziz¢2 49w“?! 06W [5’0=§), un79 He Comaé%2ma: (”0 ’V‘ X7Yé\></+:—> 4X=OJI4Y=OJ oa #(X+y)=#X+ﬂ>’=ﬂ=>X+>’éW 16> veeaPJ'xéW =>AX=07 ml ACCX)[email protected])[email protected]=>CXEW, 6' CI. CT) ‘ The Pwmd‘df‘w Luz/5 ?— I‘MWJW‘WO': CS‘LLfS‘BL (51)) (5‘7); (24)) (31), (4' 1). ' 5.6), 'Ta/‘ee. X=(4)0))V=(_170):-XJMJ 0131mm Xyﬁ‘,‘ c. T a 1a+aL4L—a a: 44 I: 4L4 = 41 L—q =444£=2z C) .2: we! 4d—c “ﬁqqc 51 4d -5 -4q+cz—+g (1-; ('1?)ch - (1%”? 50" Odmfw/Wm M'aéz‘évk 624743290444)—(1c+aal){4A—«) d~ W waxwwmm (MANN 09 veuﬂm 3. ...
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111-2007&amp;2008-1-M20-Dec2007 - KUWAIT UNIVERSITY...

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