Exam2sol - 1(6 points each Determine Whether each of the following assertions is true or false Give a brief explanation for each answer(full proof

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Unformatted text preview: 1. (6 points each) Determine Whether each of the following assertions is true or false. Give a brief explanation for each answer (full proof is not required). (a) If D: P3(R) —> R is linear and satisfies D(1) = O, D(a:) = 1, D(a:2) = 2, and D(x3) = 3, then D(f(:c)) = f’(1) for all f(x) 6 P3(R). TVWL - Tm LEM-ear (D Mel T (#003 : SIC“ W (M in bags §I,y,x‘,x‘} 4 gum, mm Mm qual. (b) There exist (non—empty) matrices A and B such that AB = I and BA = 0. mg. Same 1: by homo/ark, 4% A8) “hr (324:). Emir HUM” W ‘lV(°l=D. SIC.va Lian wees LA u at, Mith R(LA)¢§03, Mi L3 ‘5 i-l-o’l, W wlnkk ‘4‘ filly»: M4 L3 (RU-All1iol , 13w} L5(€LLA)3 = RU-QLA) , ‘0 MS ivaUeS LgLA‘fiO, (c) If {121, . . . ,vn} is a basis of V, {1111, . . . ,wn} is a basis of W, and T: V —> W is a linear transformation such that T(vi) = w, for all i, then T is an isomorphism. M. Lgl- fi: $5.”,th ’ Y': {unauwd , Tu MM): 4, T :5 it“ mid, to, knpohaséx Sm L is “awaits-mm“ T ‘5 gwggm Almkueljt T is mire $4.459. helium]: 701;) Span. W, ~42 w. as» aw amsz am Spaces wit wanna” 1‘: Warm. (d) For every vector space V, the set of linear transformations T: V —> V such that T2 = 0 is a subspace of £(V, V). For amw’kla \l‘l‘l27'J A: 3; I 13:62: . The... LA‘ :0, Sign», deLA-thHo 5am Aws: 090') M ($3513.. (e) There exists a sequence of row and column operations that transforms into OOOH OOl—‘O 0000 0000 GOOD 0 0 0 0 El};- Tu iii/5i wk» lac-s WlLBM-ltisw MSMZ. g‘fil W OMQ CVLW YrQMQ Mk . 2. (25 points) Suppose V is a finite-dimensional vector space, T: V ——> V is a linear trans— formation, and fl and 'y are ordered bases of V. Let P be the change of coordinate matrix such that P g = [33}, for all it E V. Express each of the matrices {T},7 and in terms of [T]g, P and P‘l. ‘ Y ’ WI. W Y “ {136 i P = UK“ Tl».qu [T1, :m: c [117A [T]: [11: : PM; P" [T]; = {ugh}: = Fifi/3“ A if) _ i 1' i-W ;. 3. (20 points) Find A9, Where __ cos(7r/5) ~v'Sin(7F/5-) ; : A _ (sin(7r/5) cosh/,5) ) - P TM “K (Rae/A" is “I‘M “MEX ran/s4» 4: the WM w W3“ Vs» : Tm . - ~ ' rose/,5; ‘ Shh/5') A5= m was Hem/1r- ‘ . \. W p (g I: 13 ~ > V ;§“(uA—) ’ ,Cpgfn/g) .v K; L: .51,- 4. (25 points) Invert the matrix 0 0 1 0 —3 —1 0 1 1 0 '10 ' 3 1 2" 0’“ ' You need not Show 6??er step, but you shOuld ihdicate efiough EC) We 'can"see n. o o to 1179b '3“ol 0190“, 1 000 60149.1, 3 (10 back :: e» i‘, 4x “11,: .». ~ I ‘V‘ V " '1 1-1. 5; .1 'q, s~ . ‘ w " . ,; lz': “Swf‘tk nus v - —-§ MM Wmvm4 row 1 7‘» ms '2. M4 Jm W104 “14,111 m V 197 4 m! - “111% «1,1. oo @010 ‘3; (5:28 >712 mew 2”“ 10 \000 -7.\o\ S‘mo. .5” MW» $10119, mu 3w wast b3 WlkP‘g’fi \ \ ~ : 3; '92 V gen-Dams ~ w m mi MEX, we w M M “’3 $L+wmfi~wac§ omvs Lam Knob-M? ’Ms yr-e New. ...
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This note was uploaded on 08/16/2010 for the course MATH 110 taught by Professor Gurevitch during the Fall '08 term at University of California, Berkeley.

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Exam2sol - 1(6 points each Determine Whether each of the following assertions is true or false Give a brief explanation for each answer(full proof

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