351Sp11Sec6mt2solns[1]

351Sp11Sec6mt2solns[1] - % g i mmcmwmm smamwmmmmmmmmwm...

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Unformatted text preview: % g i mmcmwmm smamwmmmmmmmmwm uawmmmxrmwvwxam E g '1‘ Q g : Math 351—0é Midterm 2 - Name: Instructions: 0 TO RECEIVE CREDIT YOU MUST SHOW ALL YOUR WORK and WRITE NEATLY. Be sure to explain your answers. 0 The format will be closed book, closed notes and no cell phones, blackberries, calculators or other such devices are allowed While taking this test. o There are problems on pages. Solve Problems 1 through 5. Choose 1 problem out of Problems 6—8 and cross out the ones that you didn’t select below. Time Allowed: 50 minutes Problems BONUS(+5 pts) Total(45pts) 2 Math 351-05 Midterm 2, March 9, 2011 Problem 1. (6' pts) MULTIPLE CHOICE QUESTIONS: (a) Consider a linear transformation T : R6 —+ R3 such that T(x) = Ax. Which one of the followings cannot be the dimension of Ker(T)? A k 0“ 3V 6'» maMy. "Mb mama) 4" 3 i é % E § g 33 g At 6 A- we. mow mat mam chm (Emmi) « B‘. 5 3% am Cmcm’j s :5 mm, a (ma be a mafia. C. 4 firm D. 3 ¢§4xm<w (A5B‘vdeACWCfQ3 :3 Q <9 2 Lesa have: ‘ Q‘XAW (WVC‘AS) :2, Q3». AW (“E-(“(1503 CQLAKd be; blgiLl’lg" (b) For which values of the constant a, is the matrix [2 g a 3 _1_ a] noninvertible? // @a=4anda=1 A “s non‘meflflo‘ “ (flth —' :2 . B.a7é4anda7él Al \ UL Me A» Q ©l‘l»q l :0 :1 3M<x 11 an/A»vamtajwmva~/umm"Wl€rmfl¢WmaW mmmmma mlWzamaimvyahvtmwlmnmmumw)wmwwmniwx‘wfiwa Wm C. a=—4anda=—1 D.a7é—4anda7é—1 W§ % (OM. 6a+c§2~120 :2: Glan‘*L5(‘:D E.a=2anda=3 Problem 2. (6 pts) Let A be an n X m matrix, and B be an m X p matrix. Show that rank(AB) S rank(A). CW“ macaw 9:. mass, Ltb gee mcmx 33> ABS? :3? For 6mm “3'34; TR?- zi> :‘i> Mafikg 7"; amcxmmexv g am (hwy :5 WKC W) § meek) emu. vkabfiw- "1mumrtvwmmmammwmmw .w. w u mflmmm mmwmx «mm ww.wemmnmww MNMmxxmwuwxwmwwwrkommwmtu Mm miwmmmmma f; a g 2 f z s Math 351~0£ Midterm 2, March 9, 2011 3 -1 3 1 W3 “4 Problem 3. (9 pts) Let A = 2 F6 —2 5 was; 3 ~6 0 8 a) Find a basis for Im(A). b) What is the dimension of Im(A)? (JAN C if“ (RE) :2“ 3) c) Find a basis for Ker(A). , W N W‘X’ \+ 3X24“ X3 W’PJXLP“: :3- ... mp. as gar-w NX%$.,D XwEE‘I [En 1&2 w. ‘ “"2" R 3;; fl: .3 :21 «L 2% A: bags. 43w W’W “‘3 { W1} X3 2: Mg: :3 0 30+ C) d) What is the dimension of Ker(A)? QM“ (Rev (/35) :1\ s 4 Math 351-05: Midterm 2, March 9, 2011 Problem 4. (9 pts) (a) Let A be an n x p matrix, and B be an p X m matrix such that Ker(A) = {0} and Ker(B) = Find Ker(AB). Please show your work. Bi:— 5’ 5mm VirfiflméO? 2%; 31 6: gm (en My; 322:0 gang wcaw §Q7l~ (b) Let the vectors v1, v2, . . . , vm in R” be the columns of a matrix A and let B be an invertible m X m matrix. If the vectors v1, v2, . . . ,vm are linearly independent, Show that columns of the matrix AB are linearly independent. ‘ a“ A N ’ x \ \1i 1 , H) \jm “er mdfl?€2\CMK”t “f W (it , H law‘s WY Cf"): "‘ 5 \ \ _. wwwmanmmmav a “\(‘N am \{QJ’ z: . 2% WW CMQ‘): $5? (cf) Qbkamng A’B m‘fi \Warifl m&ai?mwt~ g 9: i r, 3 5 g i“ S: é 8 33 é 33 S E E a e 5 § § § § g g § 2; é. E 2 i z e s § :3 E E Math 35106 Midterm 2, March 9, 2011 5 Problem 5. {9 pts) Let v1 and V2 be two unit (Hvlll : “V2” : 1) vectors in R3 such that V1 - V2 = 0. Define V3 =: v1>< V2 so that “V3” 2 1, V3 - v1 2 0 and V3 - V2 : 0.Notethatv1>< V3 : «V2, and V2 >< V3 : v1. Let T ; R3 ——> R3 be defined by “5ng \X. T(x) = x ~— 3(V1 -x)V3 - (V3 x x). “’x “ts f“? “3 “3'7 N 5“” . t . . . \iXVmV. V57“: ~““\J Fmd the B-— matrlx B of T w1th respect to the bas1s B = {V1, v2, V3}. 3‘ \ 2" Z“ ‘ u. , Mk , “3‘ “A” .33 _ flew a. «3%) , ) ‘6‘ “\(VC‘)--V‘~«~7'3>(\i»\i3\i WU? xv) "3‘ "‘3’ “ ‘ w \ ._ m“ V: “V, m» ”* ew ?’> ;?w\ ‘ 2- 35%, me [TI Va] 1‘ e m K m W Va. 35 w M l 0 6 Math 351—0g Midterm 2, March 9, 2011 CHOOSE ONLY ONE OF THE FOLLOWING PROBLEMS AND PROVIDE A SOLUTION TO THAT PROB— LEM. CROSS OUT THE ONES THAT YOU DON’T SELECT. .mw Problem 6. ( 6 pts) Consider a? anerlil linear transformation T from R" to RP and some linearly independent vectors v1,v2, . . .,vm in R”. Are the vectors T(v1),T(v2), . . .,T(vm) necessarily linearly independent? Please explain. "a «a “‘9 t “‘3” «a C-iql’jr “w‘ + C’fi‘NMXQ C“: (‘22 a r. C,“ TLC) glam” \3‘ ,H...‘\5m (GHQ “WROTE lnMpmcififilL- NY ((15% + ., . 4* CW\?“\ ‘1: T (T Wm , p in ~45 33>QVCTCVQ+ —tcw\T(\/m§ :3. C3 and mm :2 szx CD :39 T(\’7\\,,.-/ mm are \wdflj Wigwam—t. W\ Problem 7. ( 6 pts) Let 111, ug and us be perpendicular unit vectors. Show that these vectors are necessarily linearly independent. -=» » m3) , .i - Canaan" til 22—“9 ‘C a + A _\w A, A” ~> flul'ufit‘CD r, m‘ (1%” (92%“ 0 «s ._ - ~ «9 » z ‘ w v a r a “a U1 Lib C} > U” (CU-M + (LE-2flrcatia) “7" l ‘ O :w-w 2:..- ~+ e m ? ,\)\ llUl‘ll \ JEN <2th i M14 ’ > m 1%? {REC} (aorch lfi‘hli5v :12; Ciaclfifl Cab 21:».0 as a era are twig WWde Problem 8. (6 pts) Consider three linearly independent vectors v1, v2 and V3. Are the vectors v1, v1 — v2 and v1 —|— 2V2 + 3V3 linearly independent? Explain your answer. "X A , *3) ,. m3} ‘3 x ‘ ' C»\\l\“"'\‘ Clh‘jg “'l‘" (933%: D 3:5 C4 :z. C1 ': (:5: Q gm (“QT Vi 7V7“ 'V22 ONE \lnzdvg lad/Q’Fa‘dt \ \ “"1 , ,_ ~>\ Cmexcmr “3) Aimwfmfii’ J V\ 1 Ql":st and ~43 A “*5 ' t 7 A “l‘ r 3"? M -- i &\ V\ +‘Cl‘LC Vt‘“\i2> *Ql5( W “l” 2V;”’“ 5V3!) “M Q Kilt-331 4%33‘5 ere; \WQVB 39 aa— (ll-tea 3. Q 2 ’2 3 E § § E: g g E § § 3 g f i § § § mm xmvwmwa/ztwmnm»mwcmvw § § § § :3 S ;s 2 g 3 ‘ Math 351—0; Midterm 2, March 9, 2011 7 BONUS PROBLEM. (Bonus +5p‘ts) (a) Consider a 3 x 3 matrix A and a vector w E R3 such that A3w : 0 but A2W 74 0. Show that the vectors w, Aw, A2w form a basis for R3. ‘ “ '3’ 4 ' ’3’" Ms» “‘3 ml?) 9:5 f" Czkig a}: 1a€3+ C:\V~3+ C2, Pym} Ar Ca [5“ w m? ,2“ M ,1 3 2 :3 a A (befi+ 0265:1334»: w» 4» .w “w A (<:\w+Cz/\w A’Ca «=3 w? (:3 23‘3st cl Agijmg w . «a» 3 W W “‘3 'Mh L4.” :4 W (LA; w «ml/xqu -+ C5) A233 :2, Q *0 -~~.£:> «WWW V ‘ WW”; ~$ $9 7‘33 :1 Ala») uheo :1 W » «~37 w“: “N” m M @c [333% \244. 9 C?» 539. ~~ C> ~~-‘> 2 >§< 0 9 $ a M‘E‘Sfi- W ‘ m 939‘ W $0? R7)“ T<$N2 m :25 [naflmw ‘53 < C) T _ A a 7 a» A, ‘“ kw) N A; V “5‘ O W “3 [Taéxwfg Kai; C!) ‘. ) H 0 o 0 Es: Tm) “rth TEL: ~— a g a] O K §§§§§§§§§§§g§¥§§§z 95:52, £39 3 ...
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351Sp11Sec6mt2solns[1] - % g i mmcmwmm smamwmmmmmmmmwm...

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