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111-2007&2008-3-F10-August2008

111-2007&2008-3-F10-August2008 - Kuwait University...

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Unformatted text preview: Kuwait University Math 111 August. 04, 2008 Math. Sc Comp. Sci. Dept. Final Exam Time: 2 hours Calculators and Mobile Phones are NOT allowed. 1. (2+2 pts.) (a) Prove that if (Al % 0, then .4” exists. (b) Let A be an n x 71 matrix such that A3 - 3.42 = In. Show that A is nonesingular‘ 2. (2+2 pts.) (a) Let A be a 3 X 3 non-singular matrix with (1de : 7A. Show that there is no matrix B such that B2 : A. (b) Show that if X and Y are linearly dependent vectors in R3, then X x Y : D. 3. (2+2 pts.) (a) Prove that if P‘IAP and P’IBP are both diagonal matrices, then AB : BA. (b) Show that if A is diagonalizable, then A"1 is diagonalizable. 4. (3+1 pts.) Let S = {(1,2,3),(0,1,2),(+1,0,1)} be a set of vectors in R3 and X : (1, -—2, 2) be a vector in R3. (a) If possible, write X as a. linear combination of vectors in S. (b) Does S span R3? Explain. 5:.13igrPl‘fh) ___ f (a) Let X = (1,2, +2) and Y : (+2, 1, 3) be vectors in R3. Find parametric equar tions of the line passing through the point P(4, 1, 2) and orthogonal to the vectors X and Y. (b) Find the vector of length 7 in the direction of Z = (2, —2, 1) E R3. 6. (4+2 pts.) (a) Find a basis for R‘1 that includes the vectors X1:(1,0,—1,0) and X2 = (0,1,1,0). (b) Show that W : {(a, b, c) l 2a ~— b :1, o,b,c E R} is not a subspace of R3. 1 —1 Cl 2 1 —1 1 5 . 7. (6 pts.) Let A = 0 0 1 3 . Find 2 —2 l 7 (a) A basis for the row space of A. (h) A basis for the column space of A that contains only columns of A. (c) A basis for the null space of A. (d) The rank and the nullity of A. 1 U 0 1 0 0 B. (6 pts.) Let A z -—1 1 l and D = 0 2 0 . Show that A and D are +1 +2 4 U 0 3 similar by finding a non—singular matrix P such that P‘lAP 2 D. Len Roo\30‘=Kco\3Q\Q— \Q\Ir\ z) mkmoomfi= Qfimamm In :1) (Azs mnsifiuflmr mod (\f' - _}§. Gd.) (A. by \cx-;e\1\=n:n\ =>m1mA ‘51\:\ => \<—\\¢0- 2. m. \o-d'JCM : \-c:\\ —_~_-> \(‘A\1+\C>x\:o =7 \N:-x, \B\1=—\ m 193 Kat-"C z) Lax?» xffi : (_ LYX‘TW : CO : CZ) . 5. an magma} wax—«(es cmmxe) Lé‘cxmté‘i-ssfl :Lé‘aflcé‘fifl => (Mi: - ‘QGA. m P hi): D Oasdaafimfi :) K? (A?\- - 3:;PE‘PmF‘ <3" hamach =3 (\— {s omwbomjz wk. LL- d. H.131 = QQ\,’1,‘3'\ +01 £0,323 +c3 (4,0,3 \ O '\:\ \ o --l; i' :7[?_ \ 0J2 N [o \ LE—Lp, _—_>c~.m Sa\-)h‘m- 3D 7- ‘ 3 C) 0 c: j- X Gan w‘c be qkimear Codeinoskfim *5 \eguedsm uh S. 51 S' does “9‘; spam 1?}, X fié span S) KER}. ‘fl: \+ £3 3 '1 ‘5 121+5t b3 3-.- = 1 1 - —_ LL: “LL: 1 “in 3Q ’ 2}“ K3 J 3 3 '3) 0.1110“; Wflm “03 U20 0 o\ {0"}0 o\ Kc: o M\o\ Laws 0 ,I)‘ SpaniaRL-r nSSDM‘t C.{_\ o, -\‘01+C1_Qo,\,\ ,o} .x-C1Cbo a a) +CuLo,\,od) +C5Lo, aha} +QeL<-,0,G ,fl :(C,C;(';o) ~—.—_-_ Dc¢is v» (0.9;03 ci. W” 23 Wu m a may [65. l __\ O '2. ® --\ O .1. 4 [ \ _ \ \ 5 N O ‘3 ® 3 8' FM’M: (9r!) [Ck-HUF-HM o c: x 3 c2 0 0 o 7. -’L \ 1 D o o o -— k/A-\\k9\"-.sn+61 -. mum—n (7V3 '5‘. {byword , Kc): ,Ml ‘1' 9\=\ C" {L\)\)0,G\) k_1101'3)‘\1\ OA\ RRAUCQ21 ¢$ (\u\(\¥'b(_—\:'2.. ...
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