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Introduction to Linear Algebra for Science and Engineering PDF

550 Pages·2011·36.36 MB·English
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Preview Introduction to Linear Algebra for Science and Engineering

NOSRAEP NEVER LEARNING DanieNlo rman DanW olczuk • d Intruoctiotno L inear d Algebrfao rS ciencaen Engineering Student Cheap-ass Edition Takenf rom: IntroducttioLo inn eaArl gebrfao rS ciencaen dE ngineeriSnegc,o ndE dition byD anieNlo rmana ndD anW olczuk CoverA rt:C ourtesoyfP earsoLne arniSnogl utions. Takenf rom: IntroducttioLo inn eaArl gebrfao rS ciencaen dE ngineeriSnegc,o ndE dition by DanieNlo rmana ndD anW olczuk Copyright2©0 12,1 995b y PearsoEnd ucatioInn,c . Publishbeyd P earson UpperS addlRei verN,e w Jerse0y7 458 Allr ightrse serveNdo. p arto ft hibso okm ay ber eproduceidna, n yf ormo rb ya nym eans,w ithout permissiionwn ritifnrgo mt hep ublisher. Thiss pecieadli tipounb lishiendc ooperatwiiotnh P earsoLne arninSgo lutions. Allt rademarksse,r vimcaer ksr,e gistetrreadd emarkasn,dr egistesreerdv imcaer ksa ret he properotfyt heirre spectoiwvnee rsa nda reu sedh ereifno ri dentificaptuiropno seosn ly. PearsoLne arniSnogl utio5n0s1,B oylstSotnr eeStu,i t9e0 0,B ostonM,A 02116 A PearsoEnd ucatiCoonm pany www.pearsoned.com PEARSON Contents A NotteoS tudenvtis 2.3A pplicattiooS np anninagn dL inear A Notteo I nstrucvtioiris Independen9c1e SpanniPnrgo blem9s1 Chapte1r EuclideVaenc toSrp aces1 LineaIrn dependePnrcoeb lem9s5 Baseosf S ubspace9s7 1.1V ectoirnsJR 2a ndJR 3 1 2.4A pplicatoifoS nyss temosfL inear TheV ectoErq uatioofan L inien JR 2 5 Equation1s0 2 Vectoarnsd L ineisnJR 3 9 Resistor CiinrE clueicttsr ic1i0t2y 1.2V ectoirnsJR ll1 4 PlanTarru sse1s0 5 AdditiaonndS calMaurl tiplication LineaPrr ogrammin1g0 7 ofV ectoirnJRs 1 11 5 Subspace1s6 Chapte3r MatriceLsi,n eaMra ppings, SpanniSnegt asn dL ineaIrn dependen1c8e andI nverse1s1 5 SurfacienHs i gheDri mension2s4 1.3L engtahn dD otP roduct2s8 3.1O peratioonnMs a trice1s1 5 Lengtahn dD otP roducitnJRs 2 ,a ndJR 3 28 EqualiAtdyd,i tiaonnd,S calMaurl tiplication Lengtahn dD otP roduicntJR 113 1 ofM atrice1s1 5 TheS calEaqru atioofPn l anaensd TheT ranspoosfae M atrix1 20 Hyperplan3e4s An IntroducttoMi aotnr iMxu ltiplica1t2i1o n 1.4P rojectiaonndMs i nimumD istanc4e0 IdentMiattyr ix1 26 Projecti4o0n s BlocMku ltiplica1t2i7o n TheP erpendicPualrat4r 3 3.2M atriMxa ppingasn dL inear SomeP ropertoifPe rso jecti44o ns Mappings1 31 MinimumD istanc4e4 MatriMxa ppings1 31 1.5C ross-ProdauncdtV so lumes5 0 LineaMra ppings1 34 Cross-Produ5c0t s IsE verLyi neaMra ppinag M atriMxa pping?1 36 TheL engtohft heC ross-Prod5u2c t CompositiaonndLs i near Combinoaft ions SomeP robleomnsL inePsl,a nes, LineaMra ppings1 39 andD istanc5es4 3.3G eometriTcraaln sformatio1n4s3 Rotatiionnt sh eP lane1 43 RotatiTohnr ougAhn gleeA boutth eX -axis Chapte2r SystemosfL ineaErq uation6s3 3 inJR 3 145 2.1S ystemosfL ineaErq uatioannsd 3.4S peciSaulb spacfoers S ystemasn dM appings: Eliminati6o3n RankT heorem 150 TheM atriRxe presentoafta i Soyns tem SolutiSopna caen dN ullspac1e5 0 ofL ineaErq uation6s9 SolutiSoento fA x = b 152 Row Echelon Fo7r3m Rangoef L andC olumnspaocfAe 153 ConsistSeynstt emasn dU nique RowspacoefA 156 Solutio7n5s Basefso Rro w(A)C,o l(Aa)n,dN ull(A1)5 7 SomeS hortcauntdsS omeB adM oves 76 A Summaroyf F actAsb ouRta nk 162 A WordP roblem7 7 3.5I nverMsaet ricaensd I nverMsaep pings1 65 A Remarokn C omputCearl culati7o8n s A ProceduforreF inditnhgeI nverosfae 2.2R educeRdo w EcheloFno rm,R ank, Matrix1 67 andH omogeneoSuyss tems8 3 SomeF actAsb ouStq uarMea tricaensdS olutioofn s Ranko fa M atrix8 5 LineaSry stems1 68 HomogeneoLuisn ear Equat8i6o ns InverLsien eaMra pping1s 70 iii iv Contents 3.6E lementary Mat1r7i5ce s EigenvalaunedEs i genvecotfoa r s 3.7LU -Decomposit1i8o1n Matrix2 91 SolviSnygs temwsi tthh eL U-Decomposit1i8o5n FindiEnigg envecatnodrE si genvalu2e9s1 A CommenAtb ouStw appinRgo ws 187 6.2Di agonaliza2t9i9on SomeA pplicatoifoD nisa gonaliza3t0i3on Chapte4r VectoSrp aces1 93 6.3P owerosf M atricaensd t heM arkov Proces3s0 7 4.1Sp aceosfP olynomia1l9s3 SystemosfL ineDairff erenEcqeu ation3s1 2 AdditiaonndS calMaurl tiplicoaft ion TheP oweMre thoodf D etermining Polynomia1l9s3 Eigenvalu3e1s2 4.2Ve ctoSrp aces1 97 6.4Di agonalizaantdiD oinff erential VectoSrp aces1 97 Equation3s1 5 Subspace2s0 1 A PractiScoallu tiPorno cedur3e1 7 4.3B aseasn dD imension2s0 6 GenerDails cussi3o1n 7 Bases2 06 Obtainian Bga sifsr oma nA rbitrary FiniStpea nniSnegt 209 Chapte7r OrthonormBaals es3 21 Dimensio2n1 1 7.1Or thonormBaals easn dO rthogonal Extendian Lgi nearIlnyd ependent Matrice3s2 1 Subsettoa Basis2 13 OrthonorBmaasle s3 21 4.4Co ordinawtietsRh e spetcota Basis2 18 CoordinawtietsRh e spetcota nO rthonormal 4.5Ge neraLli neaMra ppings2 26 Basis3 23 4.6M atrioxfa LineaMra pping2 35 Changoef C oordinaatnedOs r thogonal TheM atrioxfL witRhe spetcott he Matrice3s2 5 BasiBs 235 A Noteo nR otatiTorna nsformaatnido ns Changoef C oordinaatnedLs i near RotatioofAn x esi nJR. 2 329 Mappings2 40 7.2Pr ojectainodnt sh eG ram-Schmidt 47. IsomorphiosfmV se ctoSrp aces2 46 Procedur3e3 3 Projectoinotnaos S ubspac3e 33 Chapte5r Determinan2t5s5 TheG ram-SchmPirdotc edur3e3 7 7.3M ethoodf L easStq uares3 42 5.1De terminainnTt esr mso fC ofactor2s5 5 OverdetermSiynsetde ms3 45 The3 x 3C ase 256 7.4I nner ProSdpuaccte s3 48 5.2El ementaRroyw O peratiaonndst he InnePrr oduScpta ces3 48 Determina2n6t 4 7. 5F ouriSeerr ie3s 54 TheD eterminaanndIt n vertibi2l7i0t y b Determinoafna t P roduc2t7 0 TheI nnePrr oduJcft ( x)gd(xx )3 54 a 5.3M atriIxn verbsyeC ofactoarnsd FouriSeerr ie3s 55 CramerR'usl e 274 CramerR'usl e 276 Chapte8r SymmetrMiact ricaensd 5.4Ar eaV,o lumea,n dt heD etermina2n8t 0 QuadratFiocr ms 363 Areaa ndt heD etermina2n8t0 TheD eterminaanndVt o lume2 83 8.1Di agonalizoaftS iyomnm etric Matrice3s6 3 TheP rinciApxailTs h eorem3 66 Chapte6r Eigenvecatonrds 8.2Qu adratFiocr ms 372 Diagonaliza2t8i9o n QuadraFtoircm s3 72 6.1E igenvalaunedEs i genvect2or8s9 ClassificaotfiQ ounasd raFtoircm s3 76 EigenvalaunedEs i genvecotfoa r s 8.3Gr aphosf Q uadratFiocr ms 380 Mapping2 89 Graphosf Q (x=) k i nJR. 3 385 Contents v 8.4A pplicatoifoQ nusa dratFiocr ms 388 9.4E igenvecitnoC rosm pleVxe ctoSrp aces4 17 388 SmalDle formations CompleCxh aracterRiosottoisfc a Real 390 418 TheI nerTteinas or Matriaxn da ReaCla nonicFaolr m TheC aseo fa 2 x 2M atrix4 20 Chapte9r CompleVxe ctoSrp aces3 95 TheC aseo fa 3 x 3M atrix4 22 9.5I nnePrr oducitnCs o mpleVxe ctoSrp aces4 25 9.1C ompleNxu mbers3 95 426 PropertoifCe osm pleIxn nePrr oducts 395 TheA rithmeotfiC co mpleNxu mbers TheC auchy-SchawnadrT zr iangle 397 TheC ompleCxo njugaatnedD ivision 426 Inequalities 398 Rootosf P olynomEiqaula tions C" 429 Orthogonailni tayn dU nitaMrayt rices 399 TheC omplePxl ane 9.6H ermitiMaant ricaensd U nitary 399 PolaFro rm Diagonaliza4t3i2o n 402 Powerasn dt heC ompleExx ponential n-tRho ots4 04 AppendiAx Answertso M id-Section 9.2S ystemwsi tCho mpleNxu mbers4 07 Exercis4e3s9 CompleNxu mberisn E lectrical Circuit 408 Equations AppendiBx AnswertsoP ractiPcreo blems 9.3V ectoSrp aceosv eCr 411 413 andC hapteQru izzes4 65 LineaMra ppingasn dS ubspaces CompleMxu ltiplicaasta i Moant rix Mapping4 15 Index 529 A Notet oS tudents LineaArl gebra-WhaItsI t? Lineaarl gebirsae ssentitahlesl tyu doyf v ectomrast,r icaensd,l inemaarp pingAsl.­ thougmha nyp iecoefsl ineaalrg ebhraav eb eens tudifeodrm anyc enturiited si,dn ot takiet csu rrefnotr mu nttihle m id-twentcieentthu Irtyi .sn owa ne xtremeilmyp ortant topiicnm athematbieccsa uosfei tasp plicattoim oann yd iffereanrte as. Mostp eoplweh oh avel earnleidn eaalrg ebarnad c alculbuesl ietvhea tth ei deas ofe lementcaarlyc ul(ussu cahs l imiatn di ntegraarlem) o red ifficutlhta tnh osoefi n­ troductloirnye aalrg ebarnad t hamto stp robleimnsc alculcuosu rsaerseh ardetrh an thosien l ineaalrg ebra coSuor,as tel se.a sbtyt hicso mparison, allgienbeirasarn ot hardS.t ilslo,m es tudenftisnl de arninlgi neaalrg ebdriaffi culItt h.i ntkw of actocrosn ­ tributtote h ed ifficulsttyu denhtasv e. Firsstt,u dednotn so ts eew hatli neaalrg ebirsga o odf orT.h iiss w hyi ti si mportant tor eatdh ea pplicatiinto hnets e xetv;e inf y oud on otu ndersttahnedmc omplettehleyy, wilgli vyeo us omes ensoefw herlei neaalrg ebfriati sn ttoh eb roadpeirc ture. Seconsdo,m es tudenmtiss takesnelemy a themataisca s c ollectoifro enc ipfeosr solvisntga ndaprrdo bleamnsd a reu ncomfortwaibtltheh ef actth alti neaalrg ebirsa "abstraacntdi "n cludae lso otf " theory." wTihlbeler n eo l ong-tepramyo ffi ns imply memorizitnhge sree cipheosw,e vecro;m putecrasr rtyh emo utf afra staenrdm orea c­ curatetlhyaa nn yh umanc anT.h atb einsga idp,r actistihnepg r oceduornes sp ecific exampliesso fteann i mportsatnettp o warmdu chm orei mportgaonatl usn:d erstand­ ingt hec onceputsse idn l ineaalrg ebtrofa o rmulaantdes olvper obleamnsdl earnitnog interptrheert e sulotfcs a lculatSiuocnhus n.d erstanrdeiqnugi ruests o c omet ot erms witsho met heorIynt. h itse xmta,n yo fo ure xamplweisl ble s malHlo.w evears,y ou workt hroutghhe seex amplekse,e pin m indt hawth eny oua ppltyh esied ealsa teyro,u mayv ery whealvlae m illivoanr iabalnedas m illieoqnu atioFnosri. n stanGcoeo,g le's PageRansky steums eas m atritxh ahta s2 5b illicoonl umnasn d2 5b illiroonw sy;o u don'wta ntt od ot habty h andW!h eny oua res olving computationaall ­problems, wayst ryt oo bserhvoew y ourw orkr elatteots h et heoryyo uh avel earned. Mathematiisuc sse fiunls om anya reabse cauistei sa bstratchtes: a meg oodi dea canu nloctkh ep robleomfsc ontreonlg ineecrisv,ei nlg ineeprhsy,s icissotcsis,ac li en­ tistasn,dm athematicoinalnbyse cautshee i dehaa sb eena bstracftreodma particular settiOnnge.t echnisqoulev measn yp robleomnsl bye caussoem eonhea se stablias hed theoorfyh owt od eawli tthh eskei ndosfp robleWmes .u sed efinititootn rsty o c apture importiadneta asn,dw eu set heoretmoss u mmariuzsee fgueln erfaaclt asb outth ek ind ofp roblewmesa res tudyiPnrgo.o fnso to nlsyh owu st haats tatemiesnt tr uteh;e cya n helups u ndersttahnesd t atemegnitv,ue s p ractuiscien igm portiadneta asn,d m akei t easiteorl earna g ivesnu bje.Ic ntp articular,s hopwru osho ofwsi deaasr et ietdo gether sow ed on oth avteo m emorizteo om anyd isconnefaccttesd. Manyo ft hec oncepitnst roduicnle idn eaalrg ebarraen aturaanlde asyb,u ts ome mays eem unnaatnudr" atle chnitcoab le"g inneDros n.o ta voitdh esaep parenmtolrye difficuildte auss;e e xamplaensd t heoretmoss eeh owt hesied eaasr ea ne ssential parotf t hes toroyf l ineaalrg ebrBay. l earnintgh e" vocabulaarndy" "g rammaorf" lineaalrg ebyroau,w ilble e quippiynogu rsewliftc ho ncepatnsdt echniqtuheamsta th­ ematicieanngsi,n eearnsds, c ientfiinsdti sn valuafboltrea cklainne gx traordirniacrhi ly varieotfpy r oblems. vi A Notet oS tudentsv ii LineaArl gebra-WhNoe eds It? Mathematicians Lineaarl gebarnadi tasp plicatairoean s su bjeocfct o ntinuriensge arLcihn.e aarl gebra isv ittaolm athematbieccsa uistpe r ovideesss entiidaelaa sn dt oolisna reaassd iverse asa bstraalcgte bdriaff,e renetqiuaalt iocnasl,c ulouffs u nctioofns se vervaalr iables, differengteioamle tfruyn,c tioannaall ysainsdn, u meriacnaall ysis. Engineers Supposyeo ub ecomae c ontreonlg ineaenrdh avet od esigonru pgraadnea utomatic contrsoyls teTmh.e systemma yb ec ontrolal mianngu factuprrioncge osrsp erhaps ana irplalnaen disnygs teYmo.u w ilplr obabsltya writt ah linemaord elo ft hes ys­ temr,e quirliinnge aalrg ebfroair t sso lutiTooni .n clufdeee dbaccokn tryoolu,sr y stem mustt akaec couonftm anym easureme(nftostr h ee xamploeft hea irplapnoes,i tion, velociptiyt,ce ht,c .a)n,di tw ilhla vteo a ssetshsi isn formatvieorrnya pidilnoy r detro determitnheec orreccotn trroels ponsAe sst.a ndapradro tf s ucah c ontrsoyls teimsa Kalman-Buficlyt ewrh,i cihs n ots om ucha p iecoefh ardwaarsea p iecoefm athemat­ icamla chinefroyrd ointgh er equirceadl culatLiionnesaa.rl gebirsaa n e ssentpiaarlt oft heK alman-Byu ficlter. Ify oub ecomae s tructeunrgailn eoerar mechaniecnagli neyeoru,m ayb ec on­ cernedw itthh ep robleomfv ibratiionsn tsr uctourrme asc hineTroyu .n dersttahned probleymo,uw ilhla vteo k nowa boueti genvaalnudees i genvecatnodhr osw t hedye ­ termitnheen ormamlo deos fo scillaEtiigoenn.v alaunedes i genvecatroesr osm eo ft he centrtaolp iicnsl ineaalrg ebra. An electreincgailn eweirln le edl ineaalrg ebtroaa nalyczier cuaintdss y stemas ; civeinlg ineweirln le edl ineaalrg ebtroad etermiinnet ernfaolr cienss tatsitcr uctures andt ou nderstparnidn ciapxaelos f s train. Ina ddititootn h esfea irslpye ciufisce se,n gineweirlsal l sfion dt hatth enye ed tok nowl ineaalrg ebtroau nderstsaynsdt eomfsd ifferential eaqnudsa otmieoa nss­ pectosft hec alculouffs u nctioofnt sw oo rm orev ariabMloerse.o vetrh,ei deaasn d techniqoufel si neaalrg ebarraec entrtaonl u merictaelc hniqfuoerss o lvipnrgo blems ofh eaatn dfl uifldo w,w hicahr em ajocro ncernisnm echaniecnagli neerAinndgt .h e ideaosfJ jneaalrg eburnad erjajdev ancteedc hniqsuuecsah s L aplatcrea nsfoarnmds Fouriaenra lysis. Physicists Lineaarl gebirsia m portiannp th ysipcasr,t floytr h er easodness criabbeodv eI.na ddi­ tioint,i se ssentiinaa plp licatsiuocnahss t hei nertteinas ionrg enerraolt atmiontgi on. Lineaarl gebirsaa na bsoluteeslsye nttioaolil n q uantupmh ysi(cwsh erfeo,re xam­ plee,n erglye vemlasy b ed etermiansee di genvaolful eisn eoapre ratoarnsdr) e lativity (wheruen derstancdhiannggo efc oordiniasto enseo ft hec entriasls ues). Lifaen dS ociSacli entists Input/oumtopduetl dse,s cribbyem da tricaerseo, f teuns edi ne conomiacnsd,s imilar ideacsa nb eu sedi nm odellipnogp ulatiwohnesr oen en eedtso k eept racokf s ub­ populations (gefnoerer xaatmipolnoesr,g, e notypeIsna )l.sl c iences, staantails­tical ysiosf d atias o fg reaitm portanacnedm, u cho ft hiasn alyussiesJs j neaalrg ebfroar; examplteh,em ethoodf l easstq uar(efso rre gressciaonnb )eu nderstionot de rmosf projectiinol nisn eaalrg ebra. viii A Notet oI nstructors Managers A manageirn i ndustwriylh la vet om aked ecisiaobnosu tth eb esatl locatoifro en­ sourceesn:o rmouasm ountosfc omputteirm aer ountdh ew orladr ed evotteodl inear programmianlgg oritthhmassto lvseu cahl locaptrioobnl eTmhse.s ames ortosft ech­ niqueuss eidn t hesael goritphlmasay r oliens omea reaosfm inem anagement. Linear algebirsea s senthiearlae s w ell. Sow hon eedlsi neaalrg ebrAal?m osetv ermya thematiceinagni,n eoerr,s cientist wil.lfi nldi neaalrg ebarnai mportaanndtu seftuolo l. Wiltlh esaep plicatbieeo xnpsl aiinnet dh ibso ok? Unfortuenlaymt,o sto ft hesaep plicatrieoqnusit roeom uchs pecialbiazcekdg round tob ei ncludienad f irst-lyienaeraa lrg ebbroao kT.o g ivyeo ua ni deoaf h ows omeo f thesceo ncepatrsea ppliaef de,w i nteresatpipnlgi catairoebn rsi efcloyv eriends ections 1.41,. 52,. 45,. 46,. 36,. 47,. 37,. 58,. 38,. 4a,n d9 .2Y.o uw ilgle tt os eem anym ore applicatoiflo innse aalrg ebirnay oufru tucroeu rses. A Notet oI nstructors Welcomteo t hes econedd itioofnI ntroducttoiL oinn eaArl gebfroar S cienacned EngineerIitnh ga.sb eena pleasutrore e viDsaen ieNlo rman'fisr setd itifoonra new generatoifos nt udenatnsdt eacheOrvse.rt hep asste veryaela rIsh ,a ver eamda ny articalnedss poketno m anyc olleagaunedss tudenatbso utth ed ifficultfiaecse bdy teachearnsdl earnoefrl si neaalrg ebIrnap .a rticuiltia swr e,l kln ownt hastt udetnytps­ icalfilnyd t hec omputatipornoabll eemass byu th aveg readti fficulitnuy n derstanding thea bstrcaocntc epatnsdt het heorIyn.s pibryet dh irse searIcd he,v elopaep de dagog­ icaalp proatchha atd drestsheesm ostc ommonp robleemnsc ountewrheednt eaching andl earnilnign eaalrg ebIrh ao.p et hayto uw ilfiln dt hiasp proatcoht eachilnign ear algebarsas uccessafsIu h la ve. Changetso t heS econEdd ition • Severwaolr ked-oeuxta mplheasv eb eena ddeda,sw elals a varieotfym id­ sectieoxne rci(sdeiss cusbseeldo w). • VectoirnsJR. 1a1r en owa lwayrse preseantsce odl umvne ctoarnsda red enoted witht hen ormavle ctosry mbo1l. V ectoirnsg enervaelc tosrp aceasr es till denotiendb oldface. • Somem aterihaalsb eenr eorganitzoae ldl oswt udenttoss e ei mportacnotn ­ ceptesa rlayn do ftewnh,i lael sgoi vinggr eatfelre xibitoli intsyt rucFtoorr s. examplteh,ec oncepotfsl ineianrd ependesnpcaen,n inagn,db aseasr en ow introduicneC dh apte1ri nJR. 11a,n ds tudenutsset hesceo ncepitnCs h apte2r s and3 sot hatth eayr ev ercyo mfortawbilteth h emb eforbee intga ugghetn eral vectsopra ces. A Notet oI nstructorisx • Them aterioanc lo mplenxu mberhsa sb eenc ollecatnedpd l aceidnC hapt9e,r att hee ndo ft het exHto.w eveirf,o ned esirietcs a,n b ed istribtuhtreodu ghout thet exatp propriately. • Theries a greateemrp hasoinst eachitnhgem athematical laanndug suiangge mathematical notation. • All-new figculreeasri llyl ustrate important coanncdea pptpsl,i ceax­amples, tions. • Thet exhta sb eerne desigtnoie mdp rovreea dability. Approacahn dO rganization Studenttysp icahlalvyel itttlreo ubwliet hc omputatiqouneaslt iobnustt, h eyo ften struggwliet ahb strcaocntc epatnsd p roofTsh.i si sp roblemabteicca usceo mputers perfortmh ec omputatiiontn hse v asmta joriotfyr eal-woarplpdl icatoifol nisn ear algebHruam.a nu sermse,a nwhimlues,ta ppltyh eth eotroyt ransfaog rimv epnr oblem intaol ineaalrg ebcroan teixntp,u tth ed atpar operalnydi, n terptrheert e suclotr rectly. Them aing oaolf t hibso oki st om ixt heorayn dc omputatitohnrso ughtohuet coursTeh.eb enefiotfst hiasp proaacrhea sf ollows: • Itp revensttsu denftrso mm istakilinnge aalrg ebarsav ereya sayn dv ercyo m­ putatieoanralliy nt hec ourasned t hebne comionvge rwhelbmyea db strcaocnt­ ceptasn dt heorliaetse r. • Ita llowimsp ortlainnte aalrg ebcroan cepttobs e d evelopaendde xtendmeodr e slowly. • Ite ncourasgteusd enttous s ec omputatipornoabll etmosh elupn dersttahned theoorfyl ineaalrg ebrraat htehra bnl indmleym oriazleg orithms. Onee xamploeft hiasp proaicsho urt reatmoefnt th ec oncepotfss pannianngd lineianrd ependeTnhceeya. r eb otihn troduicnSe edc ti1o.n2i nJR. n,w herteh ecya nb e motivatienad g eometriccoanlt eTxhte.y atrhee uns ed agaimna tfroirci enSs e ction 3.1a ndp olynomiianSl esc ti4o.n1 b,e forteh eayr efi nalelxyt endteodg enervaelc tor spaceisnS ecti4o.n2 . Thef ollowairnesg o meo thefre atuorfet sh et exto'rsg anization: • Thei deoaf l inemaarp pinigssi ntroduecaerdli yna geometriccoantle axntdi s usetdo e xplaaisnp ecotfms a trimuxl tiplicmaattiroiinxn, v ersainodnf ,e atures ofs ysteomfsl ineeaqru atioGneso.m etritcraaln sformaptrioovniisdn et uitively satisfiylilnugs traotfii mopnosr tcaonntc epts. • Topicasr eo rderteodg ivset udeanc thsa ncteow orkw itcho ncepitnas s impler settibnegf orues intgh emi na muchm orei nvolvoerad b strsaecttt iFnogre. x ­ ampleb,e forree achitnhged efiniotfia o vne ctsopra cien S ecti4o.n2 s,t udents wilhla ves eetnh e1 0v ectsopra caex iomasn dt hec oncepotfls i neianrd epen­ dencaen ds pannifnogrt hredei fferevnetc tosrpa ceasn,d t hewyi lhla veh ad somee xperieinncw eo rkinwgi tbha seasn dd imensioTnhsu.si ,n steoafdb e­ ingb ombardweidt nhe wc oncepattts h ei ntroducotfgi eonne ral vsepcatcoers , studewnitslju ls bte g eneraliczoinncge pwtist whh icthh eayr ea lreafadmyi liar.

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Norman/Wolczuk’s An Introduction to Linear Algebra for Science and Engineering has been widely respected for its unique approach, which helps students understand and apply theory and concepts by combining theory with computations and slowly bringing students to the difficult abstract concepts. Thi
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