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108AQuiz1 - id:3 9 Name u 1" Mathematics 108A Quiz 1...

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Unformatted text preview: id :3 9 Name: :- u 1'", Mathematics 108A: Quiz 1 Fliday, Api‘il 9, 2010 I’iofessor J Douglas Meme Part I. TruewFalse~ Ciicie the best enswei to each of the following questions Each question is W01 th 2 points 1. The set Z3 2 {0,1,2}, with the opeiations of addition and u'mltiplicetion modulo thiee is a field with finitely many elements. TRUE? FALSE 2 The set of vectms (123;, mg,e';3,a:4) e R‘l such that 32:; + 252 + 43:3 + 5.7.2; m 3, 3:1 + 2:32 — 5:33 + mi; = 7 is a linear subspace oi R4. TRUE (EALSEU w/ 3. Suppose that V denotes the space of all continuous functions f : R —+ 1R which have a continuous fiist derivatives at every point of 1R, 8. vectoi space over the field 1R1 0f reai numbers. Then W = {f E V: f'(.1:) = 3j(ru),for all a: E R}. is e .lillfigg';fubspece of V. @l FALSE 4. Let V be the vecto: space of tile precedhig problem Then W a {f E V : f’(m) = 3f(2:)+7,f01a11:c 6 R} is a linear subspace of V, TRUE (FALSE) 5. Let; V be the vectox space of the pieceding pioblems. Then Wm H E V:f’(0) =7} is a lineal subspace of V. eggs) Part II. Give complete proofs 0f each of the following statements 1. a (3 points) Let: W be the set of soiutéons to the homogeneous linear system :13) +232 +353 +515“ 2 0’ 1'1 +2.1”; +2213 +8.14 = 0. :L‘1 +2352 +2353 +8734 W 0' What is the coefficient mam}: of this system? 1215" l '2 I Q, 2 ‘E‘ b (4 points) Use the elementaiy 10w opemsions to put the matrix in low- 1educed (echelon fem}. in,» I, .' 7 w l ’7 Em E: E E .245 z s F 4 .2. .2»? -.| ; , ”a t 325‘ ‘5 .«fi > ) x ‘3 a. 5- “ I?“ i it; P (I; 1 X L. g 1 3 Hr E r " J 5 I ; w n f \ ’ ) ‘ i ‘ I f w. a . r '3' r”? ,f‘ ’x w .‘ i 2. w 1 ! 1 l‘ W; :3“ j (3“ 1 L3.) ' I c (3 points) Note that the solution set W is a lineai subspace of IR“ Find a basis f0] W .. . .~ .. iii: :1: ‘2 2-1:», - '3 “w :bi + 3:. 313?- 4.. .4.) A,“ k") . l a .2 : 1 PS3 + 3 r"? -"z E'- 6"?) R33 :7 9“" a. if] 7»- (“ If? r‘ "5 “ If; :(x‘ALi N31, - r‘ ‘E . 1 r“ 27 p. u“ 2‘ E if M ”" ,2 E 5* l! E E '3 4 - 1 E i " W: 4 i r”: ' ’9 i U ' 3 E % , N . L 2 k, . 3 , g . A. I N '2 n (19?“ f l (“‘3 'E '1 IE \GJQJKQJ ”J“ i O J] 3 ! V .v. o ‘I ~" 2 ' ’ ~ . j r \ > E E .\ w \ 1 ; c : Ant (J i : IO ...
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