108AMidterm1

# 108AMidterm1 - («42o Name IVIathematics 108A IVIidterm 1...

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Unformatted text preview: («42o _/ Name: IVIathematics 108A: IVIidterm 1 Apiil 23, 2010 Plait-35501 J Douglas Moon]. Part L ﬁne—False. Cizclc the. best: answel to each of the following questions Each question is wmth ‘2 points '1 Let 2.; denote aim set oi integels {0, 1,2, 3} with addition and multiplication moduio four as opcmi‘ions Then Z4 is not a ficid, because ‘2 has no unlitipiicativc invclsc TRUE FALSE '2 The set of vectols {{:E1,CI,‘3,11‘3} E [PE3 : a)? 4- 11:3 + :L'E': m 0} is o linem' subspace oil , Wm . (I, 1 4 TRUE 3 The set oivcc£015 {x 6 R41): m s(l,0,3,0) +£(0,1,1,:-1) [01 some SJ 6 M} is a subspace of ER“ FALSE :1 Suppose that V denotes the Space oi ail continuous functions f : E9 —:‘ which have a continuous Hist and second delivatives at GVGIY point of R, o vectm space ovex aim field of lenl numbels with addition and scuizn 2111:1tiplicatéon deﬁned by u mm m m.) “i am. {mm = mum). it» m e v e 3P: Then W = {f E V 2 f”(:‘c) + f(:1:) :2 0,fo1 nil 1' 6 i0} is a lilacau subspace. of V. 1/” “a TRUE j FALSE 5 ii i" is [the same vectm space consicleleci in the previous pioblem, then the iisll (f1 y}, Whch fit) 2 8": {1(1) = 3U", is lines-11hr independent; TRUE FAEEQ 6. Suppose that; W; and i-V-g ale linear Subspaccs of a vectm space V Then HG ﬂ Wg is also 11 lineal Subspace of V TRUE FALSE / T. H V is a lii'iitz-e-ciiii'mnsim'lal vectox space, the length of eveiy spanning iisi: oi vectols is less than 01 equai to the length of may iiizeazly independent list: TRUE (3%fso) Part IL IVIuItiple Choice" CllL‘lE the best HHSWBI (so each oi the following questions Each question is wcntil 2 points 1 The \rectozs (1, '2, 1) and (2, 4, t) in 3R3 me linearly ilidepel‘u‘ient. ii and onlyr ii: a 1:1 1}» t¢1 C, 2‘: = 2 c1 6% D e. None of these 2 Suppose that; v; : (1,3,U,2,ai) V? : (0:011:212)1 v; m (0,0,1,2,2), Then the subspace W of Ra spanned by the vectms (v1,v2,v;3) has climei‘isioi: a 1 i3. ‘23 C 3 cl 5 (3 None of these 3 Suppose {that T21 : 1P3" ~i 31313 be the lineal Mansioimntion defined by Tﬁx} : .r'lX, whole 1 7 5 2 Am 1 7 5 2 1 T 5 2 Then the dimension oi‘ the wage of TA is 11/3 i) 2 c: 3 (l. 4 e None oi these [\3 Part III" Give compietxz 1n oofs 01' each 01' the ibliowing statements 1 (7 points) Suppose that. (v1, :1 vectzm space V and that v1 5:3 0 Pu)er \$111211: them exists 7' E {2, [ah i: t; ,v,,.) is u. lilm'dlly dependent @152: of vecter in nu} such v, G spmﬂv], .,vjml) ‘-.‘r" “,3. _ .ﬂ, was; VI 3 , VM \‘ww ’ v1», ‘3: 2.21% CL V, "3“ - *5" ("3 J l f. A ,. x. '1' N ? J :3 V 3;; 1:: w 1?. C1,” 5 _ 1 Cum t; F ‘h in“. :1 1 J .4. ’ \ "’> 2 (7 points) ‘Néth the same condiﬁons as in {2110. pievious [noblcn], move that spmﬂvl. ,v.,'_;,v’,+1, vm) m spnn(vl, Arm) Clem), 5;; 3;}. 5;,“ i g 3W 3:); ) Nﬁidz '00 ..,1.«-'w} 49m 3 C 53”“ a “ EVZWI a . ;; E:me > )4 ﬁkan ‘3 Cpl 1:. 4 mmwﬁw Eur?) £5, Siam, M \i J 122:; m" Lily: t ,. .5? M! ' FHAA .3 : CH1: 4' ' “:M 'E' {at 2 Y Lin-4 + 01.246144 b '5" mg; n”. L K, V). w»; “,3. 7*— Laww 51,31le *3 ' " W5.“ 1 033’sij V}, ~i— \$9.313“ 4 -" 17" {<2 a [33” J {ﬁw _} _ ~ :32”) 3 (6 points) Let. T,‘ : [Pi5 w 11323 be the Eiuem tila'll'xsibzn‘mljml deﬁned by TAR) Ax, whom )75" 3750 L 4 r _- ; r-n ‘} w H'- 231 -:~ I So w W :2 p 1 , w 301-: 1qu mm - WW; m z; \ i w E ‘ J / l i 5:) “:5 \ (76 .17 7:}; :1 52:2 f ‘53 ,mN-m ...
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108AMidterm1 - («42o Name IVIathematics 108A IVIidterm 1...

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