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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'}
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2, 0\'\LV\\;\V\L 30‘». 5° L‘ hami»; «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)
nuvu = 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. Fed9Q {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 ("HA5 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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 Fall '07
 Wilson
 Math

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