111-2008&amp;2009-1-F10-January2009

111-2008&amp;amp;2009-1-F10-January2009 - Kuwait University...

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Unformatted text preview: Kuwait University Math 111 January 25, 2009 Math. & Comp. Sci. Dept. Final Exam Time: 2 hours Calculators and Mobile Phones are NOT allowed. 1. (2+3 pts.) (1 1 2 eel (b) Show that if X and Y are orthogonal unit vectors in R", then — Y” = (a) Find all values of a E R such that A = [ :| is row equivalent to 12. 2. (2+3 pts.) (a) If X = 1' + j — k and Y = 21' — k, determine sin 0, where 6 is the angle between X x Y and 2Y. (b) Determine the value of a E R such that the line x—ga = Ely—2 = z_—_41 is parallel to theplane z+2y—z+4=0. 3. (2+3 pts.) (a) Determine Whether V = {(m,y) E R2 : y > 2:72} with the operations (cc, 3;) 69 (0631/) = (I + w', y + y') and 6 906,11): (01, Cy) is a vector space. (b) Show that W = {(x, y, z) : 2x+4y +22 = 0 and 13+ 2y — 52 = 0} is a subspace of R3. 4. (2+3 pts.) (3.) Show that every ﬁnite set of vectors in R" containing the zero vector is linearly depen- dent set. (b) Show that A and AT have the same eigenvalues. 1 2 *3 1 l. 2 4 ~6 3 2 . 5. (3+1+1 pts.) Let A— _3 _6 9 ‘3 ‘2 . Find —1 —2 3 0 0 (a) A basis for the null space of A. (b) A basis for the column space of A that contains only columns of A. (c) The rank and the nullity of A. 1 2 0 6. (4+3 pts.) Let A: 2 1 0 0 0 —2 (a) Find a nonsingular matrix P such that P‘lAP is a diagonal matrix. (b) Use the diagonalization P‘lAP of the matrix A to compute A‘l, 7. (4x2 pts.) State with justiﬁcation whether the following statements are true or false: (3.) If A is a nonsingular matrix such that A2 : AT, then |A| = 1. (b) Any set of two vectors can be used to span a two dimensional subspace of R3. ((3) If {(1, 0, 2), (0, 1, ~3)} is a basis for the subspace W of R3, then W is consisting of vectors of the form (a, b, 2a — 3b). (d) If A is an n x 71 matrix such that [AI = 0, then some row of A is a linear combination of others. m\ 9 O FEM Em.“ Newark“ 95/w/aooq \. on \Q\ qto -.;1> d‘_.c\-1_—‘to :) Q6}?\7L—\,Ik. m \x -\c\\"=(><_\c3.cx-~n = \\X\\1.\—\\\T\\1 = 2 :3) \\x—~c\\—_ E21. 2.. G} X”? A. 2““ =7 Isms! : \. ‘3‘ <°)3;-"‘7'\- <\)3‘;"‘ g—L‘? Q:“°‘ '3. m \rcs m‘c quac‘nv smote, Cu,“ (-Lv‘ bu)? -\0c\,3\c,év-. b3 \N—s Va; 5°\'~«X\OO\$QQCC 98 vac 89W%WS 935% lac-w“; 4,1,1 :0 , 1+1$~§E=o =7 Subsch “n 5“ EXQ@\<\1‘9ec.6.3 ‘03 ‘17. ,gcc,9\ 1 2 ~13 o o o O O G) c: 5' ' ~ 0 o o o G) c) C.) O O C) oq {L—1,\,o,o,o), \3,o,\,a,o)‘\ woman's %v\-Qe nuﬁquce. \o\ {L\,2,J,, \\, K\,3,—1,01,Q\‘z. 1’01H c1 Raw 0. = 2 of: “mun—3 C—\ :1 ((\-\ -7_ Q E. PQWW: \')\11_Q\ : -7_ ’7‘_\:J : QQ+1)K’A+\\er\-33. o“ Q o A _‘ v‘h‘ho —'2.~7.o'o \to:o “ F:\o?‘ ’>\-:-\ [-1 —'L°,o ~[oo lzoj :>x_‘—_ l '1. 01c 2; o tic, o o o {o 0 -lo \ —3 —?_O§o ‘00, o :37: \°\ rr\:.’)_.-.[—2_. —'5 eso]~[o \o':]::>'7<-’L:[O] 0‘0 0 0 01° ,0 o olu ‘ . do 0 “’2- 0. \—\ ego ' 0-:[oazo f>\:3;['110']~[00\:0] :7K3:‘J 003 0 0 5| C) o oio 0 d 413 113 o m 9‘P~?=m =3 é‘ﬁ‘? =6’ = a“: {’6‘? - ["3 4-3 o- 0 O -”-L 1.. on \PQL\:\PF\ = \P‘\1..\<A\ :0 :3 \G\-::\ L\G\1:o) :‘Tr’ue b\ Gunw E/Kaw-Q\C '. V‘R\-R c1 {\‘ca‘1..\ «MA Keg, -31 our. :MWa—L) Lu,b,zo~-3\oﬁ:c‘ (“0,14 \D;Q o I:b + C1. C0, \,-'5\ z) [ 7. Jinan—ab ['°’ ~O\l 00. q b O J :3 5“)“: "(T‘ue on um“; 2) saw; e3 (A we. umgcngwxe/J z) Same. who is a Limr CWTMm-Y\ cg o\€tus ; \ZUQ 6WW\ ...
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111-2008&amp;amp;2009-1-F10-January2009 - Kuwait University...

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