111-2003&amp;2004-2-M20-May2004

# 111-2003&amp;2004-2-M20-May2004 - Kuwait University...

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Unformatted text preview: Kuwait University, Faculty of Science Dept. of Mathematics 85 Computer Science LINEAR ALGEBRA (Math. 111) Second Mid—Term Test Time; 75 min. ' May 9, 2004 Calculators, Mobile phones, Pagers and all other mobile communication equipments are NOT allowed. Answer the following questions: (Each subquestion is worth 2.5 points) 1.. (a) Show that if A is an n X n nonsingular matrix, then lade] 2 [Ark]. 13 41 (b) LetA: 8 g f g .Find[ade[. 0 2 5 3 2n Find a vector of length 6 that is perpendicular to the two vectors X=4i—j+3k andY=—2i+j—-2k. @ Find all values of a E R such that the angle 8 between the two vectors ' U = (a, 0,3) and V = (0,2,2) is 7r/3. 3n @ Find parametric equations of the line L through the point P(l, 2,3) and perpendicular to the plane it : 2:1: — y + 32 + 5 = 0. (13) Find the coordinates of the point of intersection of the line B with the plane 71". 4. (a) Determine whether the points (1,1,1), (2,—l,3) and {0,3,3} are on the same line. (b) Show that the set V of all ordered triples of real numbers (3,11, 2) with the operations (2:,y,z) EB (:c’,y’, z’) = (:c + y’,y +‘z’,z + 35’) and c O (3:, y, z) = (ex, cy, cz) is not a vector space. 5. (3.) Determine whether W = {(\$,y,z) E R3 : :2: + 23; + 32 = 0} is a subspace of R3. (b) Suppose that S = {X , Y, Z} is a linearly independent set of vectors in R”. Determine whether the set T = {X , 2}", 32} is linearly indepen~ dent. OO 99 0:“; 2w: Lax \gwoo‘rl .___ 0 ‘1 :017. :2(—\7=—7_ 0 1 '5 3 9. '5 *3 c: t I (. \odgcA -.—. e237”: -8 T 3 s7. 9.0a xwr= L~ "\ 3 =—\+‘J—3+‘=—Z- “xn‘f‘m's- -1. \ -1. Quad-«~03 \wyass 6 a.pr _ -221“; “2. m Gga-w'v \ \ ﬁ 6‘ \ku ' __ ‘ :> c“: 3 w WW 1 . —.—.§ “.6: .r..\L.\:o a; ‘t:'=-\ TEL pokmk‘ ergmm-dﬁm L..\_ )3,o\ , w. on Um \tawuﬁk Q\,\,\\ W‘G‘x £12")3‘ ’93 ‘z— z \5_\.“: 'E— . “We po‘mi‘ (013:9 ""3 \ *2," '2. nch m‘taa \<r_\e -.-\:—-\é§\- :3 wooinks- Oxcc‘ha‘c on PRC. €ome. \\‘J\e.. 5‘ ‘ (tn—ﬁsh mokkmosW QV‘ Drum amp ,5\ 1; “9,53 @9315} ‘0‘; \f' {s add mugg‘mr‘spcz. '_ '5- “ W” { (—11—311=5,0c\ 6R3“; - ‘ -‘w" té. me-Q‘raa colomew‘ ~— V’ C-lfd —3’2—., “5.3:.‘1 WK-LS—f'e)3‘)al\€ W {-19-1‘L'H,%\ -- {-139 351‘ , ‘5\J'3<:.\ '3 Quad} -‘3 (ch-+1“; 5495‘, 14.3%.) . 6W“ - v (434295.11 ew— M r6? Vina—7% pan—,1 : {—erBX—strﬂﬂjf 2) GW‘ I. :f Vii-2:. m’gwca. M cum 4. Clam 4&591‘) -_~ 0 -—._—_> C‘ X+L2g3w+k2mE :o, h ijyrl: Out, 2-1? C.\ :ach. 3;; :9" 1:9 CV: CL=C§ :0 o -—\_'5\$ ind-30W ...
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111-2003&amp;2004-2-M20-May2004 - Kuwait University...

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