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Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R 2 have the same size (and so does 3-space, R 3 ). That is, there is a function between R and R 2 which is one-to-one and onto. But everybody knows that there is a definite sense in which the plane is bigger than the line, and Eu- clidean space is bigger still. The line is 1-dimensional, the plane 2-dimensional, and 3-space 3-d. One of the purposes of these notes is to make this idea precise. In what follows, we assume we have fixed a field F and a vector space V over F . We start, of course, with a DEFINITION (LINEAR COMBINATION). Suppose that S V and ~v V . (So S is a set of vectors from V .) We say that ~v is a linear combination of the vectors in S if there are ~v 1 ,..., ~v k in S and scalars 1 ,..., k such that ~v = 1 ~v 1 + + k ~v k .

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McGill - MATH - 240

McGill - MATH - 240

McGill - MATH - 240

Math 240, Fall 2009, Assignment 1 due Monday September 21. Please explain yourself fully. Writing down a complete truth table does count as an explanation, but may not always be the clearest one! 1. Let p, q be propositions. Construct truth tables for the

McGill - MATH - 240

McGill - MATH - 240

Math 240, Fall 2009 Assignment 2 Due: Monday, October 5 1. Suppose that ve ones and four zeros are arranged around a circle. Lets use the word bit to mean either a zero or a one. We do the following steps: Step 1: Between any two equal bits, insert a 0 an

McGill - MATH - 240

McGill - MATH - 240

Math 240, Fall 2009 Assignment 3 Due: Monday, October 19 1. Show that if f (x) is a polynomial of degree n with real coecients, then f (x) is (xn ). 2. Prove, using induction, that 13 + 23 + 33 + + n3 = for all positive integers n. 3. Prove, using inducti

McGill - MATH - 240

McGill - MATH - 240

Math 240, Fall 2009 Assignment 4 Due: Monday, November 16 1. Your arch-enemy has started using RSA encryption, and has published the modulus n = 1919 and the public key e = 1213 on her blog. Find her private key d. Show all of your work. 2. Use a tree dia

McGill - MATH - 240

McGill - MATH - 240

Math 240, Fall 2009 Assignment 5 Due: Monday, November 30 1. (Peer reviewed question) Let fn denote the nth Fibonacci number (dened recursively by f0 = 0, f1 = 1, and fn+2 = fn+1 + fn whenever n 0). Prove that fn+1 fn1 (fn )2 = (1)n whenever n is a positi

McGill - MATH - 240

ASU - PHY - 74140

Physics 121Vectors and StatisticsName Date Section T.A. nameAbstract: The goal of this lab was to find the forces and torque in two different scenarios. In the first experiment we used pulleys at different angles with weights attached to them. The obje

University of Texas-Tyler - PHYSICS - 1444

Chapte 24. Ele r ctric Pote ntial24.1. What is Physics? 24.2. Ele ctric Pote ntial Ene rgy 24.3. Ele ctric Pote ntial 24.4. Equipote ntial S urface s 24.5. C alculating thePote ntial fromtheFie ld 24.6. Pote ntial Dueto a Point C harge 24.7. Pote ntial D

UNC - DRAM - 116

TheatreOpeningminiskit(example)o1stmoment Waitingforprofessortosayordosomethingandwonderingwhat thehellisshedoingstandingatthebackoftheroom(nottheatre noactorsorstagepresence)o2ndmoment Girlstandingandlookingatthepowerpointandwewerelookingat her(when

UNC - DRAM - 116

9110PowerPointisonBlackboard Dionysus o GreekgoddessofwineFourDionysiancelebrationswereheldeachwinterinAthens o o o o TheruralDionysiaorsmallDionysiawasattheendofDecember TheLeneainJanuary TheAnthesteriainFebruary AndtheGreatDionysiaorCityofDionysiaCit

UNC - DRAM - 116

9310Tragedy o MakesaspecialstatementabouthumanfallibilityTragicVision o Oedipus o ConcernedwithseekingmeaningandjusticeinanorderedworldAllMySons ConcernedwithhumanitshelplessprotestagainstanirrationalworldRecapfrom9110 o o Euripedes SatyrPlayTragicRe

UNC - DRAM - 116

9-8-10 The Voms o o Entrance-way on the side of the theatre Side or center entrance on the side floorThe House o o o Where the audience sits Refers to the audience Quite house, rambunctious house, etcHouse manager o o o o o Job in production ladder is t

UNC - DRAM - 116

91010RomanDramaLotsofRomanslaveswerethepeopledoingsomeofperformingincomediesand dramas PrettysimilartoGreekperiodbutdifferentinseveralwaysaswell HellenisticAge336146BCE o Movetocelebratingmilitaryvictoriesandpureentertainment o Notjustdionysiticreason

UNC - DRAM - 116

91010Legs o o o Backcurtainsonwingsofftotheside Usedformaskingbackstageareaandactorsbehindit CurtainsthathanginthewingsTeasers o Littletopshortcurtainthathangsinfronttohidthegrids PhotoCall ReadThrough o o o o Oneoffirstthingthathappensinrehearsal Jus

UNC - DRAM - 116

91510 KNOWQUOTESWELL FRIDAY o o o o Lookatspecificquestions Readplays ReviewTheParchmanHour o ByMikeWiley Wantedtocomeandtalktousabouthisproduction Aboutthefreedomriders Putintoaprisoncalledparchmanprisonwhenarrestedonthewayto D.C.PopQuiz o Thefirstch

UNC - DRAM - 116

91710 Renaissance Between14thand15thcenturies,Italiandramabegantoreplacereligious,mystery,andmorality playswithseculardrama. TheybasedthisworkonRomanplays,whichbasedtheirworkonGreekplays NeoclassicaldramabasedonAristotlesPoetics Demandedverisimilitude(str

UNC - DRAM - 116

92910(UPONBLACKBOARD)PaperAssignment? o o CometoclassMondaytogettheassignment? Isthepapernotduethen? OthelloActIIISceneI o o o o o Jammedpackedwithevents Takesplace Onhoneymoonevening IagoswifeEmiliasetsupatimeforDesdemonaandCassiototalk EmiliainformsCa

UNC - DRAM - 116

Drama83010PRIVILIEGECARDASAP Calendarhasbeenupdated Officenumberandsyllabuschanges(tweaked) READINFOLEADINGuptoplaysandINFOAFTERassignedplays POPQUIZonRisingoftheMoon o 1)FuntionaryPolicmentXandB(whatrolethecharactersplayed) o Themovementcentersaroun

UNC - DRAM - 116

E lements of Theatre for Each PlayOedipus Movie o Really dramatized and showed the usage of masks Plot o Oedipus is determined to solve the mystery of who murdered the King Laios. This will end the plague in the city of Thebes. o In the end he discovers

UNC - DRAM - 116

EverymanPlot o Everymanistoldofhisdeathtocomeandwantstorepent o JourneytowardsdeathandattempttoavoidhellandbecleaninGods eyes o Everymaniseventuallycalledbeforegodandthusshouldlooktoendof hislifeevenashebeginsit initialsinsatyoungagewilleventuallycauseso

UNC - DRAM - 116

EXAM ONE Drama 116o The Actor Actors job Believable behavior under given circumstances Actors tools o Voice o Impulses o Body o Concentration o Imagination Warm-Up o Vocal o Body o Imagination Actors union The Union for professional actors, stage manager

UNC - DRAM - 116

Drama List on blackboard The voms-entrances on ground floor (right left & center) The house = audience (how big is the house? How quiet?) House manager= manages the theatre (lishtens to how audience reacts) Late comers= people who come in late (let in & p

UNC - DRAM - 116

DramaExamIStudyGuideTheActor/Audience Theatre o Performancethatplaceshumanexperiencebeforeagroupofpeoplean audienceinthepresentmoment Twogroupspresentfortheatretohappen o Actors&audience Whatmakestheatreunique o Itsaliveness o Livingartforminpresenttime

UNC - DRAM - 116

Oedipus(9/3/10) Movie o Reallydramatizedandshowedtheusageofmasks o MoremovementseenthanintraditionalGreeksetting PlotoOedipusisdeterminedtosolvethemysteryofwhomurderedKingLaios. ThiswillendtheplagueinthecityofThebes.o Intheendhediscoversthatheisthemurd

UNC - DRAM - 116

OthelloPlot o Pathoftragichero,Othello,andgoingfromhappyinlovetoamurderer TragicflawofjealousyandIagosmanipulationo Severalsubplotsrevolvingaroundothercharacter Exposition o BeginsinstreetsofVenice Conflict o IfaffairistrueornotoIagowantingpowerandpla

UNC - DRAM - 116

The Rising of the Moon(ROM) by Lady Gregory Plot o Sergeant, two policeman (X and B) and ragged ballad singer are main characters o Posting up placards, one gets put on placard near port o Looking to catch an infamous fugitive for 100 pounds and promotion

UNC - ECON - 410

9210 Testtwoweeksfromnow o Someshortanswerquestionsandproblems,maybesomeMCbutneverbeforeLastclassdiscussedframeworktoincorporatepreferences Utilityfxnseasiertosolveproblemsandwewillincorporatepoliciesusingutilityfxns S1: o o o o DidnttalkyetaboutwhatisFE

UNC - ECON - 410

Chpt 3General rule for optimization: o Choose the IC as far from the origin as possible while not exceeding budgetUsual application of the general rule: o Look for bundle that sets IC and budget tangent (equal slopes) and exhausts all $The Usual rule i

UNC - ECON - 410

91410 Exam Likeoldexamformat Calculator Listentorecording Donthavetosimplifyeverythinglikesquarerootof39forexample Canwritedirectlyontheexam Bothmathematicalandconceptionalandgraphical DontworryaboutMRT(notabigdeal) About6questionsindividuallyweighed Sees

UNC - ECON - 410

9-21-10S2 o Objective o Max profitChoices Outputs Input How they are going to produce them Labor Capital . What goods and services How much/ quantities of each outputS3 o Consumers have constraints Technology What is feasible to produceBudget (-like)

UNC - ECON - 410

92310S15 o Isoplot Whatsconstantalongisoplot OutputorproductionS16 o o Isoplotisjustastraightline PerfectsubstitutesS17 o o PerfectCompliments Lshaped/rightangleisoplotsS18 o o Productivityofinputs MarginalandAverageProductsofLabor Table L:between07un

UNC - HIST - 213

91710CurtissDesigntheJN4Jenny o o Usedoftenastrainersduringthewar LindhbergalsousedthistoperfecthissoloflyingFirstAeroSquandron o LookingforPonchoViaPonchoVia o PunitiveexpeditionwasmilitaryoperationconductedbyUSArmyagainstforces ofMexicaninsurgentFran

UNC - HIST - 213

91010 Bombingfromtheairpanickedpplbutwasntasbadasfirstthoughtfinallydiscovered Firstselfpropelledballoon o JulesHenriGiffard ThaddeusLowesroute Stillnumberoneprobleminairshipstodayis_ o WhereyourgoingtomoveshiptoThaddeusLowe o AmericanCivilWarandranU

UNC - HIST - 213

91310 THESISPAPER VETinternetsourcesinbibliography(vettingthesourcesestablishwhythissourceis credible) o CanusesourcenotentirelycredibleifitisusedinacertainwayUseaparagraphforeachdifferencepoint Trytogetunbiasedairforcewebsitenotobjectivebutcanhavegoodst

UNC - HIST - 213

9-15-10Thesis paper o o o o o o o o o o Sentence 1 2 or 6whatever it takes How to vent the sources Note if ppl have certain interests for the reason they have their info Justify sources and their reliability Determine the bias and recognize it and be awa

UNC - HIST - 213

91910Machinesin1914 o Aircraft o Typicaldesignwasmonoorbiplane Averageweightwhenempty10002000pounds Topspeeds5585mph Ranges100250miles Ceilings700010000feetAirships 400700feetlong Topspeedof2550mph Enduranceofupto100hrs Ceilingsofupto20,000feetPilotsin

UNC - HIST - 213

92210Movies6p.m.inHanesArtCenter121 o INCENTIVE o o o o Questionsonmidtermandexamconcerningthesefilms29SeptTheSpiritofSaintLouis 13OctoberTheBattleofBritain 27OctoberTwelveOClockHigh 10NovTheFogofWarCommandoftheAir o o Discussedbytheoristsbeforewar Fin

UNC - HIST - 213

10110 Nextmovieintwoweeks TheeyesoftheFleet o o o Ashortseaplane Makefleetmoreeffective Tellnavalofficersalreadyfamiliarwithneedforscoutingandreconnaissance andconvincethemthatwarshipscouldbemadebetter o Replaceshipswithaircraftsthataremuchfasterandseef

UNC - HIST - 213

HistoryofAirPower KnowledgeisGood Discusseddesiredlearningobjectives o o 4)DevelopaSenseofHistoricalDifference. EnjoytheHistoryofAirPowerandAviationTechnology(ThomasEdison&) o Anythingintendedtomodifytheenvironmentforthebenefitofhumanityorsome subsetofhu

UNC - AFAM - 101

83110 HowU.S.justifiesslavery o Democratic? o o Believedindemocracyandnothierarchyandhereditary(kingsetc) Societybasedonslaveryforfirst100yrsofcountryshistoryMostpresidentshadslavesbeforeslaveryended JohnLocke InvestorinslavetradingcompanyinGreatBritain

UNC - AFAM - 101

9210 QuizTuesday o o o Chpt1 LastThursday NewmaterialfromtodayJustificationsofslaveryandAfrica DonthavetomemorieseverysocietyinAfricawelookat Inbook o o o Reviewquestionsatendofchpts Broadquestionsfromtext Ifsomethingisinaboxonpgintxtbkdontskipbcitsallim

UNC - AFAM - 101

Cleopatrablack? o o Dontknowifshewasblackwhiteormixed. Doesnthaveanythingtodowithitbackin30BCbcracedidntmatterthenfor herReligionfromAfrica o o o Voodoo Catholicreligions AllofNewWorldnotjustNorthAmerica(likeIansweredgobackandlookup)LanguagefromAfricaa

UNC - AFAM - 101

9-7-10Papers are now due on the 28th o o o o o Topic is think topic Discuss legacies of slavery in contemporary society Specific prompt is included Can use I in your papers Your opinion but explain and justify with examples etc.Movie Thursday o o o Begi

UNC - AFAM - 101

9910 Move:TerribleTransformation ComplementaryNotesBASICTHEME(recap) o o Indenturedservantswasfirstmethod Whynotjustgetwhitesfromoverseas Freedomduesexpensive Bchadtobegivenfreedomduesandfreedomyouhavepopulation competingwithyouintheend o Twofreedomduesb

UNC - AFAM - 101

92110 Virginia Didntlegalizeslaveryuntil20yrsafterMassachussetts(eventhoughfirstcolony) o VAtobacco,CapitalismWhenNCwasestablishedtheinstitutionofslaveryalreadywrittenintoSalemWitchTrials Tituba WasfirstportrayedasaNativeAmericanandlatershewasportrayeda

UNC - AFAM - 101

93010DeclarationofIndependence o o o o o Explanationofwhywewereallowedtobreakaway Unalienablerights Naturalorder 1776 LanguagerheteroicandideaologyConstitution o CreatedlawsofAmericaandgovernmentTuesday o BringinstudyguideThursday o o Finalexam DSS

UNC - AFAM - 101

SlavesatUNC UNCwasphysicallyconstructedbyenslavedppl OldEast,thewallaroundcampus,Southbldngbuiltbyslaves ElishaMitchell o o o LargestslaveowneratUNC(about1000people) Feltunderpaidsorentedslavestostudentsanduniversity Arguedslaverywassystemofpropertyhold

UNC - AFAM - 101

92410 MoviesandDates FinishingWWI Airgroundsupport o Early1916alliedsuccessrateregardingsupportmissionswaslowbutpotential wasobvious o AlliedexperiencesledtomoresophisticateddoctrinesBasicdoctrines o Putenoughaircraftinareasofigherscouldsealoffbattlear

UNC - SPAN - 101

Study Guide for Exam 1. Span 101 Online (Leccin 1- Leccin 3)The Exam 1 is a 50-minute paper exam that you will take during the class hour with the instructor instead of having virtual class. (NOT ONLINE!) Check your exam schedule and come to the class to

UNC - SPAN - 101

F inal Exam Study Guide (translated)Context: Carlos, an exchange student, meets Tina at a basketball game and invites her to dinner. I. Before leaving.a.Carlos and Tina are nervous. They dont know what clothes to wear. You are going to read a list and

UNC - SPAN - 101

Alt+control+shift+?= Control+i,a,o= Control+shift+~key+n=

UNC - SPAN - 101

How to type foreign symbols To get foreign symbols in Windows applications, you can always pull down the "Insert" menu. However the best way to type foreign symbols such as accents, , , , is changing your keyboard setting to United States International. T

UNC - SPAN - 101

Argentinemencangreetwithalightkiss. Inmostbusinesssettings,peoplegreetoneanotherbyshakinghands. Theygreetwithonekissoneachcheek. Theypatoneanotherontherightforearmorshoulder. Theywouldprobablyshakehands.

UNC - SPAN - 101

Leccion1EschucharDr.Cavazos: BuenosdasSeorita.YosoyelDr.AlexandroCavazos.maletas.S,seorita.S.Dos.Muchas graciasseorita.Gracias,adis. Srta.Martinez: BuenosdasSeor.AlexandroCavazos,ce,a,ve,a,zeta,o,ese.Cuntasmaletasdeun usted?Tres?Bueno.Nohaydequ,Dr.Cavazo

UNC - SPAN - 101

List of Answers1.Muy bien, gracias. Me llamo Christopher Edwin Carr. Soy de Clinton en Carolina del Norte. Soy bajo y rubio pero soy un joven trabajador. Tengo veinte aos. Mi cumpleaos es el doce de junio.2.3. Hay _ estudiantes en mi clase de espaol.