Midterm 1 Solutions

Midterm 1 Solutions - Last Name Special Notes: Math 211:...

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Unformatted text preview: Last Name Special Notes: Math 211: Linear Algebra Midterm 1 Spring 2011 Solvt/l’t'ms First Name Time: 90 min. a) Textbooks, notes, notecards, calculators, cell phones, electronic devices, etc. are NOT allowed. b) To obtain credit you need to show all your work. Justify all of your claims and arguments fully, and write clearly and legibly. c) Use the backs of the pages if you need extra space. Honor Code Statement: Sign after completing your exam. Problem 2 Problem 3 Problem 4 Problem 5 I have neither given nor received aid on this examination. Signature M 1m?) l W Aolvvl‘t'avul {Mamb— W3 as oi ross'l’v 59/33/va (Law/“cot Solw’l’l‘w dW zed/b" ’ProlaLe/vw . ' (1) Suppose that 1 -2 —1 2 c a A_[*2 5] and AB=[6 _9 d], andlet B—[bl b2 13.3]. (a) Explicitly find 51 and 132. (b) Suppose that 133 = b1 + b2. (i) Is the matrix equation [51 132] x = 53 consistent? (ii) Explicitly find 0 and d. (a) max Aim/1L AB # [Ma 45; A53} . Wore/(ova was sae, —’ A a 2L ’ _ 0 Mt Abi: [co] . M #9; [fall 3/14/1143 +w MHX 41> Jirolvg Wm (4W 'JOMUIA. x/LCM. ‘- 1a2._.1_fl_)l#2,*1__9[Io’7] WW6 [r169 oI‘I ol 4 its] '—-'> [(1 F wlxw‘of/L WIN“ (lo) a) H3 33: Smog) We cam Ware. 103 M a fiwr o I; M F; Duh/(00k Wm 444i, M1. la I mm. 'X [T3] '62]? —, [:3 r5 Wl.s4,%b3 ma Varhw/ULV) it?) 1.5 a Mtwhm‘ ATS—5‘ b3 AbE'M-hm of MAJWCX MrH'Pl/Cmqh'm’ I; 2 a SL1 pose A iS a 3 X 3 matrix and that b is a fiXBd vector in R3. the matrix equation ( p AX —“ b has a unique solution, explain Why the columns 0f must Span R3. LP A14 3x3 Moi Af=g (AM a L/golwfi-W) WM 4.0% REEF (tummy/div? 41> Mm Jamar axiom {5 [Li cl 2 0 x ) magma 9/? L9 . (Hove % warramiz M3 magi o 1 MWbflr.» WWW“, Mm cal/Mimic be a flaw looks like [wow MM IKE—16F, MA go MIG rs Cows‘si'wt «Coy _ gem? “Mg WM.) Moi 0.445 "fie/“23 mm £36 va‘Hm (Li a or; W0 mt W CoibLmM Pr Jéo 14:41) MJ‘M"['TWL) 4%: (gs/{9 0/? A SFaIAIR. ( b) If A is a 3 X 2 matrix, explain Why the columns of A cannot possébfiy span R3. W A ['5 3X2) M {+5 Column/5 (We ’2. Vac—Furs m I123. I'F We oomu'ofiw JM: A2}: 0i all WWW XII/Luv Wmh‘m a{ Z vac/I’WS) JIM 7/ Moors Xx”; Me @ Vim. m [£13 ) Mafl s: Malay Mt'M'OM Wi'H (mafia M /&f)€£o€ Piawl in. [R . L? We Jake alt/fl vac-hair whim 0,946.»; PMM’ We Wm M vw wag Mama ft m a. “away omega. of 4m MS 0M. 0 ' Consider the followin diagram depictin veotors in R2 without labeled coordinates. g g g (1) How many solutions does the matrix equation [51 52] x = b have? (ii) How many solutions does the matrix equation [51 52 5'3] x = 5 have? (4;) W a Wt MAJ/W. ,{A out xCQaS‘L 1 Solw'i’l'm him-LAM. We swim/5 kaelK At. Ififil+WJa=G, W «HM/re. Weraa ‘the “find; \raVL'abLL )imowm/ov, m wank/L OOW'MFD'MQ Jm) ’Fw Hawk) «fa/w fi'nug M M 9011.44! M xgt’bw’t 0/? 2:1 boo/~40 Hex/uwa Ni W M M MN: is at Fara/Ltd!) chm WI'HA Sl‘fltfl sz-Ifi’lfxfld~ V" gate MUN/fl Mfol'w’c mm M MM 04.21: wl/vose au‘agmafie M E. (Simiav’b for k3, «Wei am “15:2. WI’JHN 4C1 Q4211.) Mia/{WCfJ/We. t‘s a, mabm Salt/chow. Q1) W MMX £§bmrifiom rs omisvi'mt GLA Thaw/£4- l'lA' 0—)) M 50 since. [3, 31%} 1'5 2X3, 3JW4, t‘s mama/Mij a {ii/tea Vania/bu (ad, MS’L Z Yu‘WJrS ToSél'bU). mm; are 0% mam/v soil/L5. .25- may, Am 4 I —2 (3) (a) Find all values of b for which [33] E R3 lies in Span ( |:—2:| , |: 7 :l ) . 1 4 —3 Bonus: Interpret our answer geometrically! é, SFCLIA ( @ «Ru/we kaS‘t Sada-IVS 0(1) 04; 6i? Sft. ' ,1 4 F3 ' 9(‘20‘. 24' __ chili Ar '. 2 d 5m (:5 = M e awn—t ‘ 4 '3 I 4.11-3“ l aillgdz, \ 44.3 a Wx'si'mut fiwv‘_ Mae/m. Mgr @M—JDYMW) "1_2_ c} I'Z 4 {*2 Ar 1.52. q- }..1 4f 4. '1 is *9 o 3 we ——> o 3 W43 —> 0 0 run —> a [-3 "r :3 | o 3’ ~15 0 I —3 0 4’ #5 o o .wi’l m Wile/m {5 camera“: e> lumt mu [‘5 WA; 015 mews [ooléol Melba: we see We. W b+|1: 0 4w .1. Wésl’m‘t 1mm, W &) ice’l’] «EA me WW 5. W (b) Determine whether or not Bl),(_11 30,6 is aspan— 3 3 - I ning set for R2”. If so, write _4 1 as a linear combination of these matrices. If not, find an explicit example of a matrix that is not in the span of these matrices. I: i] . . 2‘52, L542. 1| 0- 5 Cu mvtj s ' @r—‘a Q W W 1 I o , 1m wide A 0L F ab :E far [i3 [K 51% (HELD)? #3 (Mug WK [fidlélfi cam in wuLliu/LM a/{QWV W0_ at W Ar mam/CW _ er.» lam mi‘st Sew/(avg £1,301: ) 9L3 , A4 a HZ ,H; [titm[iél+d1[‘?‘il*'4s[?iiWm e [:2 = r: rt [:gfi 4:333] 4' [:4 :43 g 3; 5L to] = “1434914 “I'd?- C d 0C2}? ’&3+9{+ {:3 d1“°¢5"‘”‘*4 = a J) 6L WtS’i/t/w’t Amer O< ’OLz 7' b . fl ! m sling: c M”, ' M91“? 6! J — 9L3+o£4= A ... (3b) M‘WK .. (Some. jinx/{as Me, Wfrwm a) a. I 0/! l “a bra. F?) O | 'l l 3-40 __.) low— 0 0 0 i +u¢aL ,d o o 1 '1 an" we. mph-0J6 rFrv WLM at: awi) A; @4071) o(3= [0*LEM’L'JR mg ~ a—a - 3 3 5L" C" .okq-EHA'laL .mwc, (ifan [:4 I] Wu Iota phi) _ H ,. «I: 2 _ X323 50 Wk 1?}1-[LU4/cafi Uflfifi'jfl‘rfi) (4) Prove any two of the three following true statements. (You may prove all three state- ments for extra 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.) MW? P‘Ha) Recall that O[—1, 1] is the set of all continuous functions from [—1,1] to R. Then G 5' = {f E Cl-1:1]If(—1)= 0 and f(1) = 0} is a subSpace of C‘[—1, 1]. (b) If A is an m x to matrix, B is an n X r matrix, and o: E R, then MAB) = A(ocB). (c) If A and C are n x n matrices such that CA 2 I, then the homogeneous matrix equation Ax = 6 has only the trivial solutiOn. Co») 5 4:. man} to M‘wdh'm at dLgmfidflmecwh o we E4, [3 gah'gg'w £(,[):0 MAL L(l)T—O)A So 4;! am MAW/6L or? a {t/L Wz/Qwe, g «LA (60 Let, £66 5 M aLellZ. M We. WSJC claw/water ob? M ’PYDVZQ. 0L? e go W9. coMfac/whz. («em = We) Moi mm W I " ab 0 emu £és =9 2PC’I)='O M ELEM/{Lug = O wawofoeui. '"' fi'erX so 00? QRJDSXTa M 4b 5:020 in M mum/cf" vi 5; t‘e- M63. (LL) Lat {:65 mistakes. We mtemveM£+fiaeS. MIA/l1. (Ptfibyt'l) = £(‘O 4’ 53(4) I up vacA-ov amqfi‘m O 4, O sine-e S —:.' o 13"”?“43 '34 Dan?— ' glmlM/Lj) (duh l) : Wefire) Li»? 6' g . =— o 0,) Wm B: [13,-33 Wt- ream. mg M A15: [A5] INow CWx‘aLa-r J 0&(A’B)= W136] AVng afld' 0’? AB 2 [MAW “0:591 d6? MC 3mm [Mu/ML. = [mam mm] _ A‘mwg a? mutthfi = Mod; Jr] Ad. o¥ WMULXWM. r. Mag-:5“, 3,1) M. 0% sea/w Mat. 7' M Mfr-6A. (a) SWWM M CA=I amt 32mm»; m3, Wlwh'tm Rib-#5. Mm, we. (CA)? 5 IS? C-(Aifi Is: my. oqf moo 0% Math CE’IQ WFDLU-Mfis 045- ASE—‘5 a): £32 Damn-ma; :5 4 c :3. 2;] 0 x FmVMJjQ oat 1 Mxkfé, "$2- 5) so 3 (S W le 5631th JP? A3? = ‘5 0,) ma: La) m MAL m WFm-M cg ‘ Win/£34 Wlh‘Vfllc ‘m W'9W&&. (m, [39”??? 50 aim/r cw). Jugs 4m, gag/ML dog/LL 92:6,- : NWW?W'M 0/? a My a Cofwwum, \LaC/{TY M, M Msjrract/Ouéim’h'm” rub WM . (5) The following statements are either true or false. If true, prove them. If false, provide an explicit counterexample which demonstrates that the statement cannot be true. (a) Suppose that 37 and 2? are solutions to the matrix equation Ax = 13. Then 37 + i is also a solution to the matrix equation Ax = b. KAI/66: (tics 0. mm 01C floats/vahwg r’oizaAJJI/éor: Mgflvt- fame. Cautiwlej- ALL-5m)=M+A2; =b+b52b Ml ohm/weal M i/g w + s) xii-(Wit) s b. ) (Em/«23mm A= Li] M w [313- ‘M W— at“ C..WDS£ '03:. 75,: [2'1 §MCL 5E. HDWflA/W) 2. Algae)" = [:l¢[:3=l:) So (3+5: (1)) Let V be the set R2 with the following scalar multiplication and addition; w [:3] = [iii and [:1] + [:33] = [31:13:] - Then V forms a vector space with these operations. 1594,65 = (Ym om 23ml; Ala/Uh Wat): fist-M3 4am. m myim) lami plow/we. Mt juice Matt'th A M some. M M Mm}; aAaUQi’l'm/L M £1) UH) ~ (A4) will wivme ' MUL go 480% mfg/(MC have, 3+M’l'tit Will/{A— UVS) Ame; > ' x (MUM/LA mtijaL= L,ch 9L: 2)€=3)M [‘2‘ 1 l [1' mm {H.3le s (mil-1"] : a. , ’ ((0 “Eu/’6 Z'fg'lnl): Zl-Sal [1-5j:[@] Ami Asia [1,] 9e [if] , (A?) W mt x6wa (he all alselfl M_afifl [122. Q (6%) M60 > low/UL (is) M (Ma) mm. 5 ...
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Midterm 1 Solutions - Last Name Special Notes: Math 211:...

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