Midterm 2 Part 1 Solutions

Midterm 2 Part 1 Solutions - Math 211: Linear Algebra...

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Unformatted text preview: Math 211: Linear Algebra Midterm 2 Part 1 Spring 2011 Time: 90 min. Last Name 5 0‘ S First Name Special Notes: a) Textbooks, notes, notecarde= calculators, cell phones, electronic devices, etc. are NOT permitted. b) To obtain full credit you need to SllOW all your work. Justify all of your cieims and arguments snflicienth, and write clearly and legibly. c) Use the backs of the pages if you need extra space. (c1) Always look for the most efficient argument. Do not do unnecessary work! Problem 1 Problem 2 Problem 3 Problem 5 Total Honor Code Statement: Sign after completing; yorir exam. I have neither given nor received aid on this examination. Signature ' 0 3 —6 (1)Let-A= 10 0 . 0 4. —7 (3,) Write A as a product of elementary matrices. (b) Use part to commute dct(A). 5 1 " ”‘ '" 5m. E“wa+£é:‘>mté“>- Cb) (£40k ( k) 52 E3 54) d‘ dauqv 1 2 (2) Let u = 3 and v = 5 . 1 . 0 (a) Find a, vector of the form w = [U] such that {u,v:w} is linearly independent. |_. 1. (b) With the choice of w you made in (a), let A = [u v w], and compute det A. (c) Give two different reasons that A must- be invertible, and find A‘l. (0,) CMl-OLBV Jim bu:ijme maulw'x Heabmfll-n'm [ ‘5 o l 0 I20} [10 17—0 12.0 i 3§0-—735‘0 -—>)o-'ID _=,o[o___> [[1 v [04 r o—«l r 00 r 0 may 0 ) W a Iva val/in €25 r: 0. flows/F:er 33,0”);573 1.1: Ah. 0 “NJ-’10- ‘g r‘f‘ 0- WI; mose- $=[03 ’gw Cavevmizxvtfifl. 1 2. o 1 0°» Exiwwtal dbl: [3 g a] 0‘4va Colwmw 3 l l l MA: o—o+l.owc['lj= sup : #1 2. s” l 160 l 3 1 o 0 r (C) A J25 t'nvmlw'lal-L A’s/1a out A 5 -l WE 0 ) emu? /X?l'mC/L .MJUL column/5 0/; A M [/Lsmj M Same. Amy/twee 94 WW QFM'W ms in (a) We em mm wine A4 lay «SlM/li'vtfl wi‘J/h I: 1 2 —1 l 2 4 *2 2 (3) LetA— _1 _2 3 1 . 1 2 1 3 (:1) Find bases for CUIEA), Row(A), and Null(A). (b) Determine whether or not A is invertilfle. 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Howe/vex)I % A32=fp M Wis’X’MA/JC, JHAJ. ’Rf-QL vmwu {up Aifib’ MAO WM MAME AUX-2T; wl'H Amt/e A Ra “mafia ML 00 mag golwk'm. flue/M, A3211; m mm 0 ‘5‘" °° WW3 golwh'm/l . [4) Define. :1 0 m2 —2 X1: 1 , X2: —2 , X3: 0 , X4: —8 . —1 1 1 5 (:1) Compute the dimension of Spa11(x1}x;,X3,X4), and describe it geometrically. (1)) Suppose that a : x1 + 2X2, and define A = [a x1 X2 X3]. Compute rai'1k(A) and 111111ity(A). (c) Find a basis 5 for R3 containing as many vectors from {3:1, x2, X3, X4} as possible. [(1) Find the transition matrix which changes from 5-coordinates to B-ooordinatos, where 8 = {eh 92: 63} is the standard basis for R3, and B is your basis from (c). (6:) Label your basis vectors 5’ = {b1,b2,b3}. If x = (1, 2, 3)T, find scalars shrugs, such that x = clbl + (52b2, + (:3b3. . , - *2- 1 o -L -L cw,th l o '2 =- 1 o 2 (a) 1 «2 0 —s a i 4 o 4: a on— L w "l l 1 S 0 l '1 3 D l "l 3 - ~ r2. .9 l 0 *1- 2‘ ' ° 2' IW'fi/LL mam Mt o ’1 xi 3 0 o 0 0 X5 Mi )gf a/ri/ .A- 12% {TE—fa fit'mr OM05 9F 321"" X: . mew, tot/um rm M M? stauta’fimgfi) = (i!) 35;), Mohfi avg”) “‘ch coiuWLfl/S i'* 1 Dr? W42- gun—Jail??? Coiwwm is? l 0L W WWW-«Q ave) mag SD X15522 owe, Jim. {MUE- Moi W ‘L bat/$43 {or 41AM, NM Siaqu‘yvfitp4) = 2-. . l’an-QMI‘WMQ MSW“ 04 “13 Vela a Pia-M .JrflthQ-flmjii‘“. (to) ran/Lt (£0 = dimCeflCA>) Mot lag (a) J 733 6 SPM(§i,)'Z-D. AlSO, 3T=§fi2ig mam/M aeSFaM‘ihiz). wage) S?““(aiii)§ji 72,3) 5 SW” (gi)?z>' Aaim'“) m'mce' 521‘“); are film. n‘AM. ‘03 (0‘)} wWCKflA) = Z. New, [HA—v Rank~NULU|3 Wm xii/um 3 4i wls : 4 = rk(k)«i Miii'bCA) '5 ZJr MWUA'i’DCfi) => autism) =2. Use: this page if you need mom space for (4), but make sun: you me not; doing unnec- essary work! {That warming appflfilcs cguaifly to (15$ probiems.) (C) gum [L1 = .3 W «since m9 5;" >2; ave Liva- min? KFWWL Uh) we MM'L $93,322:; Jfiv M's. wig MM. Wm?)me 0,? Am MJ‘VLK W WHX JAM Jau- finale? Colwmm, 3 fiw.;'kj1f. \JeC/Jrfirs [m “23 £01m 0L fia-M'S L3 BMI'S/n/LMS, m we my w «Cw i=[EE'b531-[iJCB m Mao [sag dam; l o o l o o l o o l o o 0 —-l 0 4/2. V2. 0 4 0 l o Vz'yz 0 0 '1 l 7-; 1/1 I 9 O I V2,. Ya. I I o o _ __ _ MOE J30 {Th-72,0} J45 qu. rR-rmthm Y1. 71. 1 L6) Mk6 (’07 [32363, ayLvezw 711M, 0 o ‘ 1 b3 (i), [i3LBZEIL‘E/z. = “Hm WM 7L Vb l 2’ I (5) Prove any two of the three £0110wing true statements. (You may prove all three state- ments for (swim credit, but clearly indicate which two you would like graded for regular credit! If not indicated otherwise, I will assume that the first two proofs you write are regular full credit submissions. independent of correctness.) (a) Let A and B be n x n matrices. Prove that if B is singular, then AB is singular. (b) Let x1, . . . .xk be linearly independent vectors in R”, and let A be a nonsingular n x n matrix. Define yt- = Axr for t’ = 1, . . la. Prove that y1F . . .,yk are linearly independent. (e) Let A be an m x n matrix. Prove that if B is a nonsingular m X m matrix, then BA and A have the same null space and rank. M 3 04 W WW6 HW (Ft/blow? // (lo) 54% HWO: 3.3.11 Cc.) See MW 8'. Bio. ZZéL ...
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Midterm 2 Part 1 Solutions - Math 211: Linear Algebra...

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