
nunayude sidhaantham

ganithashaasthrathil, prathyekichu lye aaljibrakalude sidhaantham, lies sidhaantham prasthaavikkunnath, swabhaavasavisheshathakalaaya poojyatthinte beejaganithaparamaayi adacha feeldil, formula _ 1 ennathu pariharikkaavunna lye aljibrayude parimitha-dimensional praathinidhyamaanenkil, formula _ 4 ulla formula _ 3-nte invariyant subspaceukalude oru flag formula _ 2 undu, athaayath oro formulakkum formula _ 6, "i" ennivaykku aaa formula _ 5 ennaanu.
mattoru vidhathil paranjaal, soothravaakyathile ellah rekheeya parivartthanangalum mukalile thrikonaakrithiyilulla matrixukalaal prathinidheekarikkunna vidhathil "v" nuu oru adisthaanamundennu sidhaantham parayunnu. ithu frobeeniyasinte falamaayundaakunna oru saamaanyavalkkaranamaanu, kaaranam yaatra cheiyunna maatriksukal oresamayam mukalile thrikonaakrithiyilullavayanna, kaaranam yaatra cheiyunna maatriksukal oru abelian lye aljibra srishtikkunnu, ithu oru fortiery parihaaramaan.
0 enna swabhaavasavisheshathayude oru feeldil pariharikkaavunna eathoru parimitha dimensional lye aljibraykkum oru nilpottant deriwade algibrayundu ennathaanu lies sidhaanthathinte anantharafalam (#consequences kaanuka). koodaathe, oru parimitha-dimensional vekttar space "v" leeii oro flaggin, oru borrell subalgibrayundu (athil flagine sthirappeduthunna rekheeya parivartthanangal adangiyirikkunnu); athinaal, formula _ 3 formula _ 9-nte chila borrell subalgibrayil adangiyirikkunnuvennu sidhaantham parayunnu.
ethir-udaaharanam.
p"> 0 enna swabhaavasavisheshathakalude beejaganithaparamaayi adacha feeldukalkku, praathinidhyatthinte alavu "p" nekkaal kurravaanengil (chuvadeyulla thelivu kaanuka), ennaal alavu "p" yude praathinidyangalil paraajayappedaam. 1, x, d,/dx enniva vyaapichukidakkunna trimana nilpottant lye beejaganitam oru udaaharanam nalkunnu, ithu "p"-dimensional vekttar space "k" ["x"] ("x" p ") ll pravartthikkunnu, athinu egenvectorukalilla. eee trimana lye beejaganithatthinte ardhadishaya uthppannam "p"-dimensional praathinidhyam (abelian lye beejaganithamaayi kanakkaakkappedunnu) edukkunnathu pariharikkaavunna oru lye beejaganitam nalkunnu, athil ninnu uruthirinja beejaganitam nilpottant allah.
thelivu.
formula _ 10-nte alavilulla induction vazhiyaan thelivu, athil niravadhi ghattangalund. (kurippu thelivukalude ghadana eamgalinte sidhaanthavumaayi valare saamyamullathaan.) adisthaana cases nissaaramaan, formula _ 10 nte alavu positteev aanennu njangal anumaanikkunnu. v" poojyamallennum nammal anumaanikkunnu. laalithyathinaayi nammal formula _ 12 ezhuthunnu.
ghattam 1: sidhaantham eee prasthaavanaykk thulyamaanennu nireekshikkuka
theerchayaayum, sidhaantham prathyekichum parayunnath, formula _ 3 leeii ellah rekheeya parivarthanangalkkum poojyamallaatha vekttar spanning formula _ 14 oru pothu eejanvektaraanennu. neremarichu, "v" oru saadhaarana eejanvektaraanenkil, formula _ 14 athinte spaanilekk edukkuka, thudarnnu formula _ 3 kwaandant formula _ 18 ll oru pothu eejanvektar amgeekarikkunnu; aargyument aavarthikkuka.
ghattam 2: formula 10-ll codimension onninte anuyojyamaaya formula _ 19 kandethuka.
soothravaakyam _ 21 ennathu uruthirinja beejaganithamayirikkatte. formula _ 10 pariharikkavunnathum positteev alavullathumaaya formula _ 23 aayathinal, kvosant formula _ 24 oru poojyamallaatha abelian lye aljibrayaanu, athil theerchayaayum codimension onninte ideal adangiyirikkunnu, koodaathe ideal correspondence anusarichu, ithu formula _ 10 leeii codimension onninte idealumaayi poruthappedunnu.
ghattam 3: formula 27-ll chila leanier functional formula _ 26 nilavilundu, athaayath
poojyamallaathathaanu. ithu inductive sidhaanthathil ninnu pinthudarunnu (eejan moolyangal oru leanier functional nirnnayikkunnundo ennu parisodhikkaan eluppamaanu).
ghattam 4: formula _ 29 oru formula _ 10-invariyant subspace aanu. (eee ghattam oru pothuvaaya vastutha theliyikkunnu, athil pariharakshamatha ulppedunnilla.)
formula _ 31, formula _ 32, appol nammal formula _ 33 theliyikkendathundu. formula _ 34 aanenkil athu vyakthamaan, athinaal formula _ 35 ennu karuthuka, aavarthichulla formula _ 36 sajjamaakkuka. formula _ 37 umm formula _ 38 umm ettavum valuthayirikkatte, formula _ 39 leanier aayi swathanthramaanu. appol avar "yuu" srishtikkunnuvennu nammal theliyikkum, athinaal formula _ 40 "yuu" yude adisthaanamaanu. vaasthavathil, ithu anganeyallennu vairudhyathiloode anumaanikkuka, formula 41 ettavum cheruthaanennu karuthuka, appol formula 43. formula 44 rekheeyamaayi aashrayikkunnathinaal, formula 45 ennathu formula 39-nte oru rekheeya samyojanamaanu. mapp formula 47 prayogikkunnathu formula 48 formula 49-nte oru rekheeya samyojanamaanu. kaaranam "m"-nte minimaality anusarichu eee vektarukal oronnum formula 39-nte oru rekheeya samyojanamaanu, athinaal formula 48 aanu, namukk aavasyamulla vairudhyam labikkum. oro formulakkum formulakkum formula 53 num adisthaana feeldinte formula 54 polulla gadakangal undennu induction vazhi nammal theliyikkum.
formula _ 57 cases formula _ 58 muthal nereyaanu. ippol nammal chila formula _ 52-nteyum formula _ 19-nte ellah gadakangaludeyum avakaashavaadam theliyichuvennu karuthuka. formula _ 19 oru ideal aayathinal, athu formula _ 63 aanu, athinaal
thudarnnu induction ghattam pinthudarunnu. ithu soochippikkunnathu oro formulakkum _ 53 enna subspace "u" ennathu "x"-nte oru invariyant subspace aanu, koodaathe adisthaana formula _ 67-leeii niyantritha mapp formula _ 66-nte matricks formula _ 68-nuu thulyamaaya diagonal gadakangalulla mukalile thrikonaakrithiyilullathaanu, athinaal formula _ 69. ithu "x"-nuu pakaram formula _ 63 upayogichu prayogikkunnathu formula _ 71 nalkunnu. maruvashathu, "u" vyakthamayum "y"-yude oru invariyant subspace aanu, athinaal
commutaterukalkku poojyamaaya trace ullathinaal, athinaal formula _ 73. formula _ 74 invertible aayathinal (bays feeldinte swabhaavathekkurichulla anumaanam kaaranam), formula _ 75 umm
angane formula _ 33.
ghattam 5: oru saadhaarana eejanvektar kandethi thelivu poortthiyaakkuka.
l" oru ekamaana vekttar subspace aayirikkumbol soothravaakyam _ 78 ezhuthuka. adisthaana feeld beejaganithaparamaayi adanjirikkunnathinaal, "l"-nte chila (athinaal oro) poojyamallaatha moolakathinum _ 29 enna soothravaakyathil oru eejanvektar nilavilundu. formula _ 19-nte oro gadakathinum aaa vekttar eejanvektar aayathinal, thelivu poortthiyaayi. soothravaakyam _ 81
anantharafalangal.
eee sidhaantham prathyekamaayi oru (parimitha-dimensional) pariharayogyamaaya lye aljibra formula 10-nte anubandha praathinidhya soothravaakyamaaya 82-nuu swabhaavasavisheshathakal poojyatthinte beejaganithaparamaayi adacha feeldil baadhakamaan; athinaal, aaa soothravaakyathil oraalkk oru adisthaanam thiranjedukkaam. oro soothravaakyathinum 86 enna soothravaakyathinu poojyangal adangiya konukalundennu eluppathil pinthudarunnu; i.iii.formula _ 88 karsanamaayi mukalile thrikonaakrithiyilulla maatriksaanu. ithu soochippikkunnathu formula _ 89 oru nilpottant lye beejaganithamaanu. koodaathe, adisthaana feeld beejaganithaparamaayi adachittillengil, adisthaana feeld athinte beejaganita klosarilekku vyaapippikkunnathiloode oru mithya beejaganithatthinte pariharakshamathayeyum nilpottansiyeyum baadikkilla. athinaal, oraal prasthavana avasaanippikkunnu (mattoru ardham vyakthamaan):
nunayude sidhaantham kaarttante parihaarakshamathayude maanadandatthil oru disha sthaapikkunnu
vaasthavathil, mukalil paranjathupole, adisthaana feeld vipuleekarichathinushesham, implication formula _ 97 eluppathil kaananakum. (neremarichu theliyikkaan kooduthal budhimuttaanu.)
nunayude sidhaantham (vividha "vii" kku) eee prasthaavanaykk thulyamaanu
vaasthavathil, nunayude sidhaantham eee prasthavanaye vyakthamaayi soochippikkunnu. neremarichu, prasthavana sariyaanennu karuthuka. oru parimitha-dimensional formula nalkiyaal, formula 101 oru paramavadhi formulayayirikkatte-90-submodule (ithu alavukalude parimithathayaal nilanilkkunnu). appol, paramavadhi, soothravaakyam _ 103 lalithamaanu; athinaal, ekamaanamaanu. induction ippol thelivu poortthiyaakkunnu.
oru abelian lye aljibrayude parimitha-dimensional lalithamaaya module oru dimensional aanennu prasthavana prathyekichum parayunnu; eee saahacharyathil oro vekttar subspace oru lye subaljibra aayathinal eathu bays feeldilum eee vastutha sariyaanu.
valare upayogapradamaaya mattoru aaplikkeshan ithaa
lies sidhaantham anusarichu, namukk 105 enna soothravaakyathinte 26 enna leanier functional formula kandethaam, angane 105 enna soothravaakyathinte 29 enna bharasthaana soothravaakyam undaakunnu. lies sidhaanthathinte thelivu 4-aamm ghattam anusarichu, formula 29 oru 10-module aanu; athinaal formula 115. prathyekichum, 116 enna oro soothravaakyathinum, formula 117. formula 89-ll aprathyakshamaakunna 10-leeii oru leanier functionalilekku formula 26 vyaapippikkuka; formula 26 appol formula 10-nte oru dimensional praathinidyamaanu. ippol, formula 123. formula 124, formula 105-leeii formula 26-maayi poruthappedunnathinaal, namukk aaa formula 127 undu, athu 105 enna soothravaakyathil nissaaramaan, athinaal formula 107-nte (lalithamaaya) praathinidhyatthinte niyanthranamaan.