108AMidterm2

108AMidterm2 - Name: ]'<(Jr V J NIathen‘satics 108A:...

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Unformatted text preview: Name: ]'<(Jr V J NIathen‘satics 108A: IVIidtei'm 2 May 14, 2010 1310;138:301 J Douglas Moon: Part I” True-Faisal. Circle the best unswel to each of H19 f'ollowiz'lg questians Each question is worzh ‘2 points 1 The Image of the lineal map T : €32 ——‘r R2, deﬁned by MEWS 3) (115:)- is the stmégilt line Llnough {the 011gin spanned by the vectm (2, —l) ‘2. The null space of the lineal map T 2 E1333 w \$13-32, deﬁned by 3;; m] 1 0 2 T :13 “(a 1 "'2 has a basis {(—-2, M3, 1)) L” TRﬁ® FALSE 3 The nonhomogeneous linem system (111.13] -£- (112.13 + -+ agnm'n : bl, 021151 + (122% 4* +(I211-Tra = 52, (1111731 4"“ 3:113:52 '" ’ ‘ + (111113511 ""3 bm has a solution f01 any choice 01 (1);,53, .. ,bu) it and oniy if the cm‘lesponding homogelwous linear system (“1:51 —i~ {112312 + + mum" = U, {121\$} + (1.22:1); + - + (LgnfL'n :: 0, {1,113,} + {Ln-351:2 + ~ —I- an” 1'“ m {J has :10 solutions FALSE ;1 Let @193 be the set of infinite sequences {:111,:v3,. t,:1:,-, ‘ ), \vhele each rt,» is a real mnnhel The function T : R” —-+ 1P:m deﬁned by T(:1:1,:1:g,:r:3, ): (0,:r1,[},rz:g,0,:1:3, .. ) is a smieetive lineal map JHJWW TRUE "' F ALSQ 5. Let W“ he the set of inﬁnite sequences (34,114,. . .,:2:,-, .. ), \vheie each IL“; is a lead mnnbel. The linear map T : IFL’O" w R00 deﬁned by TH], \$3,125, m ((33,:33,:1:.;, has a nonzelo nuil space spanned by the vectol (1,0, 0, ) 'l/W—un- TRUE} FALSE 6 Suppose that V denotes the space of nil CGlltilll‘IOUS iunctéons f : 3F: m» 1?: which have a continuous ﬂush and seeoz‘ld dezivntives at ever point of £91, a vector space evex the ﬁeld EFF-i of 18211 numbels with addition and scuhu muitiplicntion deﬁned by (f + QM) = f0) 4" Wt (MW) 2* 6141(0)» f0: fth E V and a E Then u! = {f e V : f”(t) — [(t) m me: an t e {9:} has 3 basis eel'lsisting of the functions (6", e“) FALSE Part II. Multiple Choice. Cizcle the best answer to each of the following questioz‘ls 7 Suppose that LA : 1R5 m> E1333 be the lineal tzanslblz‘nntion deﬁned by L..1{x) m Ax, whele E ——4 {I U ——1 .41 = 0 0 1 U ——5 O 0 O 1 m3 Then the dimension of N(L,l) is a 1 f b 2 c 3 d :1 e None ofthese n 1 h 2 e d 4 e, None of these [0 9 Suppose that L4 : 1P:2 wr L”? be the lineal {nunslbamation deﬁned by I.__.‘(x) m AX, whole ,1 m ﬁ/z w1/2 ‘ 1/2 ﬁ/e Then L ,1 1ep1esen£5 u eounl'eleloclnvise 1c>tnl2ion tluough what angle"? f... a 17/0 I)“ 7r/4 c 7r/3 d 7r/2 e 10 If T : R5 ——+ lR3 is a lineal t1 unsionnution such alum; the dimension 01' RH") is two, then the dimension of N(T) is f 3) d. 4 Part III. Give complete details lox each of the following p1oblems :1. 1 i). 2 (2 None of these 1‘ (6 points) Let; P208) denote the space of polynomials of deglee thlee, wiah basis 7 m (pmphpg), whole *1 p0(:y)m1, j)1(.’1£)m1II. ])g(.?l) = a. , and give E313 the stanclmci basis 13 = (81,83, ea) Suppose the: T : 7320131) m» IE3 is the lineal L1 ansf'o: motion deﬁned by TOM 1)) = ' :2». i' w e .N l 2 ‘ :13 f 2__ e”) "3, J to L l M 5 o a -. l; i m- r») F I! 6 g I x. / None of t;l'iese '2 (1053051115) :1 Complete the following sentence: A list of vectois (V; , V3, Jr") in if is iinemly independent: 11 and only if -;.. - 7 G“ rf- -‘ ‘ him {1% V... i I i i r 1) Suppose that T : V ma W is an lujective line-ax map, and that (vhvg, ‘ ,v,,) is n liuezuly independent list of vectozs in V Piove that; (Th/*1), T(Vg), ,T(v,,)} is a lii‘iomly independent: llSl‘. oi vectms in W Hint: Shut. by assuming (L1T(V1)+ + an.T{vn} : k ---‘:9 I e m- "x J .... "WM i “ f i ’ w '2: ' i E 9 t- 0 w gm l ('3 E3 ‘n 13“ C l r A zr‘ ‘ 3 i 3r ". 7" i W 71. ail: r: L 1‘}; (“1%fo n; : "3;, " l4 l J by -- ~= x MFR :. 4’ :1” LA x), l: W/ a \\/1 i i or: a i ‘ x”. ‘ ‘ a"? "'1, ll , a: , . Hint: Finis]: by showing that. (11mm; x -- \$11,150 3 ( 8 peints) The Main Tllemem £10m Cllapi; n1 2 in the text ls: Theorem. If V is a ﬁniL‘e dimensional vector space and T : V Mr W is a lineal map into a ‘-’CCf§01 space H", the” dim V m dim NU“) + (li1‘1‘2R(T) Recall the idea behind the pwol We stall; by choosing a basis (1111 ‘ ,um) lo} N(T)H An Extension then states that. we can extend this {:0 a basis (“13' sunhvlw sVn) of V If we cam show that (T(v]), ,T(v,,)) is a basis lbl [{(T), Chen cli11‘1R(T) m n. It. will the“ fellow that dim V m“ m. -i— n. 1": dim NC?) + Clim RH"), and the themem will be pmven Thus we need only Show ahaz; (T(v1), u ,T(vn)) is lineally inclependem and spans MT). vae that: elm list {T(v1), . ,T(vn)) spans R(T) (You do not. need to Show £zl'mt (T(V1), . ,T(vn)) is linemly independent ) :1: :ﬁf‘r v! 7“: l l... x; 3 ~ l 1»- '~—‘? 2 x ' ‘ J < I}, V .3“ {1‘1 'L’l % i C" “I l L‘ f \ll 2 I' 5."; " :' J. a E l;=,.f’m. r”: "a ’4 f" ’ “1:1”. , ‘ 1 A m I My, " f ‘ w" \ ' "1 l . 3 1:2,} (alum, ~ r m: 7 u - ~ . i 2”! v. E , l ( ‘ I I _4l l rm : f * ‘f w u {,1. l1 ll 5' m ‘ﬁ L " 4 4:“ '2’ r” #1: I "-‘.‘J Wu» "‘""' m m l r W h l { v 3 M" w J I I m i ‘r f 1 W I r I I v x Lax ~ [ 'g x I a. ‘ ‘ 1 I f \,j i j 'V W 5 , -- l “’ v~~ I l“! i l I ...
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This note was uploaded on 06/05/2010 for the course MATH Math 108a taught by Professor Moore during the Spring '10 term at UCSB.

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108AMidterm2 - Name: ]'<(Jr V J NIathen‘satics 108A:...

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