math208midt2solns - MIDTERM EXAM 2 Math 208 Fall 2007 Dr...

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Unformatted text preview: MIDTERM EXAM 2 Math 208 Fall 2007 Dr. Robin Wilson Name: §SOLUTION$2 I o Show all your work! Answers with no work will be given no credit. 0 Check now to be sure your exam has 7 problems on 8 pages (including this page). 0 Don’t forget to check your work. a Good luck!! 2 (16 points) — _— “16) — WW — —— ( 7 10 points) Total (100 points) 1. fints) Let A and B be 3 X 3 matrices with det(A) = 4, det(B) = ——3. Use properties of determinants to compute: (a) det(AB) : 1+ -3 :E (b) det(BT) :E] (c)det(5A) : 531:4 : \2S.k\ cho} (d) det(A3) : @A/flg = “(3 : (e) det(A‘1) : ..‘_, t .x 2.( 16 points) Determine whether or not the following sets of vectors are linearly independent: (a) {(—2, 0, 1), (3, 2 ,5), (7, 0 ,2)} ‘. 2 7» 5mm dx‘itO CHM 92*“. (b) {2—m+4m2,3+6x+2x2,2+10x——4x2,1—m2} ‘mead &In&1y\\' 591C! 0 bflJU 9* R tux} “[7 3) t‘ew‘nb‘ 53" W‘§ 34' M Lf .dewu W": .2 3 Z \ o -| (o 10 o 0'} ,, u m o o N. 7. 3 2. a o L‘ 7- _.1 _‘ O H 2 §Ll .1 o .4 ‘C; [o o (3 ~l (a [O 0 O 0 7,1 ‘ 0 ~ 0 ‘ $15 ea 0 o 23 3C0 I O O l _3_g_ -I 28 3;; o | (,3 ~10 o 0 O \ 3'32 ——‘- ,5 ,5 o w ,. - eondwb; M :é‘sfskvvx dam, MJF o o y y 0 Mg, ~ 2, 0\'\LV\\;\V\L 30‘». 5° L‘ ham-i»; «JU, “@er —. 3. (16 points) Determine Whether or not {the set of all n x n matrices A such that AT = — is subspace; of the vector space Mnxn. ‘V A} 0 V! 0 Suwox RT?~A cud ‘ZBTg'fg. new ' “(va 4w? =, -9 +43 .- -(Pr+B> 7—) Suffox' C:5 fi SL&U( «A AT: — A (CA? = em = c(-M —- ~(CA) 4. (16 points) Consider the subspace W of R3 where W = {(2a — 3b, b, a + 4b) | a and b are in R}. (a) Findabasis for W. (ma—3b) b J <1th) = (zmo) a3 + (33),) 1%qu = 0(z,o,\) + Manet) §(Z)O\n)('3\l)l'ng \N 0 Lung,“ h/ (b) What is the dimension of W. dim W912 (0) Give a geometric description of this subspace. 0‘ [$9wa 444 %L 0&4 m‘an.‘ 5. (10 points) Given that u = (1, —2, 2,0) and v = (2, —1,0,2) find the following: (a)u0V 3: “1* _2. _‘+ 3‘0; 0.2 : a *9; fl “W” um: 2’+(4Y"r(0i“r(2)I = “Hf“ 4—61. :- (c)d<u,v> MUM z uu—vu .. HA): (\-Z)'2-(-0\Z'0)0'7“3 :(_\"5’2"Z) nu-vu = m) (E \L _ 6. (16 points) The following matrices are row equivalent: ‘3 6 —-1 1 —-7 1 —2 0 —1 3 A: 1—2 23—1,B=0 01 2—2 4" 2 —4 5 8 —4 0 O 0 i T (a) Find a basis for the column space of A. (d) Find the rank and nullity of A. M (YavikA— Z Viuivfiw Pf”; 3 7. (10 points) Determine whether the following statements are true or false. Justify your answer. 2- 3 7 (a) R isasubspace ofR. Fed-9Q {Q J, U; “at (”new 0 N36 ‘4» 0C [323! (b) The nullspace of A is the solution sef of the equation A3: = 0. w ("HA-5 Sm CM). d2€~m§hw Qt nullsfaw) (c) )The columns of an n x n invertible matrix form a basis for R". @111 (ll: ““4 ‘4 lMMYHlolI OYLMM 011*qu AMA rams moQ COMM“ n; owe a \gas“, Qfi YR“) (d) det(A + B) - —)det(A + det(B) 3ij :xsi . [2; :2] Wm _. M 0mg" ) 0M $= ml dela>rd4w22...
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