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final - Math 107 Linear Algebra and Differential Equations...

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Unformatted text preview: Math 107: Linear Algebra and Differential Equations Practice final exam Name: : Friday, December 11, 2009 Lecture section: 107.0 Recitation section: 107R.0 All answers must be justified. No calculator is allowed. Question 1. Find conditions on a, b, c so that n = (a, b, c) in 8&3 belongs to W = Span('u1, n2, n3), where u, = (1,2,0), in = (—1,1,2), in = (3,0, —4) Set V= xw+_._:yuz + 2013 a (ab, c) = mam) + 9‘“““”* H 3’0! “”49 -_-.- (x43), 3%, 2x+ g, 21y- 4?) 229‘ =3 helm» 'fb Pecke (ab'fvrn/l: o 6 ~61 vaA 0 0 Di lie-26 +3C Question 2. Find the general solutionI-df; the system ' :5’1 2 2532 + 2:03 2:5 2 2561 + 2:173 333 = 2$1 + 2562 chew my is z del{~A .—2)\ zip-:0 =,> 6X+Z)C)\:‘*)=° 2 2 - /\'-"Z , ‘+ m A=‘t v.=- [I] ._ - '0 EVAE'Z V;=Lé] ,.V3"'4l‘§ _3..j ' “I . I _, 0 .25' pm»): q[-(e t+ (Janie 2r+c3[’]e l Question 3. Find a basis and dimension of the subspace W of R3 where (”W {mm- WW: 05 0,23%!» 5.; dual/11(3) #1933 Lflg‘f'bfiu a be PM var 7h a+b+ cw owe/33M ' .. —0 N-L '{DVO Mepehaficm‘ VCG‘IDM — J M2 -— I mg 0mm var->2 , ow! MUM; firm “Laws gem Question 4. Solve the initial value problem m’ = Am with So. x (-(L) —~ I'Ie’2t(6o§5t “301513). - 2 '5 .. ' #5 m 09190115 2 f. ~2t PC0397 e t Question 5. Let V m P(t), the vector space of real polynomials. Determine Whether or not W is a subspace of V. Explain. (a) W consists of all polynomials with integer coefficients. (‘0) W consists of all polynomials with degree 2 6 and the zero polynomial. No. 179“”): 7'36 few £64): -—'{:‘g+f5 Luv} 71;a.)..£=e) 1‘5?” (c) W consists of all polynomials with only even powers of t. ((95 (fiww dwell?) Question 6. (a) Find the matrix of fundamental solutions for the homogeneous system [ZiHéiHEi :4 =4 -=-> 9cm =(zet x- =><+g¢x+<a€t 9M6): Cater 46161: t tat m = e 0 e k (b) Find the general solution for the nonhomogeneous system mt— - [31] 1 t ”t ”t { I -1 [e {'8 6’ "(Me 9*“ " [o :tyfelflé 9* 2/ €641 = ttct-I-iefj mil 9" KG):- Eeffifite‘f: tat-tie? , Question 7. (a) Find an invertible matrix P and a diagonal matrix D such that A=PDP-1 A: 6 ‘2] 0M __ ’ [6 —1 Final (grammes ¢ agewb" °T ”<- _ -— =2 )i: 213. , A3?“ 3331 3 . J\=3, v=[§] so Wig g} M P-jz if (b) Find A"? Without directly inverting A. A4 :(PDP'DVI -'-' Pb P Question 8. Let S consist of the following vectors in 5R4: ‘d k u1=(1,1,0,—1), u2=(1,2,1,3), u3=(1,1,—9,2), u4(16,—13,1,3) .. (s) Show that 8 is orthogonal and o basis of m4. 0? gag {1154: 1“ COMM“ “I'MZ =(+2+0-3=- (./-__ 0 ice . virus-1 " “'0 WM?” ’0 M2 Hit: ' ‘0 M! “Y"? :0 WWW“; .. -_- WM, 3 is who ) A'OGOVAIX-g, S 23:3)0’53 ‘9” (Pl? MW (My W .. tr Q/Wj (Mcpméém'f’ l/(a‘vrj 13m. (2/1345 /5 of» (R . (b) Find the coordinates of an arbitrary vector v = (a, 11,0, 0!) in 3%4 relative to the basis 8. C}: 3239': aiiSZi Mg“, 3 b +c+3ql c .-.— ALL/‘3. = L... 2 M2““‘2 (5 . (6q_(§b+C-('3d C4 - L—ufiru = (753'— ...
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