Table Of ContentNOSRAEP
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 .
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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.
