
resheey kalavadhi

ganitat, vectorchya sanchacha (vector spacemadhun) resheey kalavadhi (jyala resheey patwar kinwa fakt kalavadhi asehi mhanatat), darshavila jaato, too vectorchya serve resheey sanyojanancha sanch mhanoon paribhashit keela jaato. 
udaharanarth, done resheey swatantra vahak eka pratalavar pasaratat.
resheey kalavadhi ektar serve resheey upasthananche chhedanbindu mhanoon kinwa samavisht asaleli sarvat lahan upasthan mhanoon darshavila jaau shakato. tyamule vectorchya sanchacha resheey kalavadhi haah swatah eka vector space asato. spanche matroids aanee modulesmadhye samanyikaran kele jaau shakate.
vector space haah upasanchacha resheey kalavadhi aahe heey vyakt karnyasathi, samanyatah khaleel vakye vaparali jaatat-ektarah spans, haah eka spanning sanch aahe, jyadvare pasarlela/tayar kelela aahe, kinwa generator kinwa generator sanch aahe.
vyakhya.
ekhadya kshetravar vector jaaga dilyas, vectorchya sanchacha kalavadhi (maryadit asane aavashyak nahee) haah tyateel serve upasthanancha chhed mhanoon paribhashit keela jaato. tyala 'spann buy' kinwa tyateel vektardvare upkshetra mhanoon sambodhale jaate. yaulat, tyala 'spanning seet' mhanatat aanee aapan mhanato qee 'spans'.
vaikalpikaritya, 'chaa kalavadhi haah ghatakanchya (vector) serve maryadit resheey sanyojanancha sanch mhanoon paribhashit keela jaau shakato, joou vareel vyakhyepasun anusarato.
sootra _ 1
anant, anant resheey sanyoganchya babatit (i.eae. jethe ekhadya sanyojanamadhye anant berries samavisht asoo shakate, asey gruheet dharoon qee asha berries kasha prakare paribhashit kelya aahet, samaja, eka banach space) vyakhyedvare vagalalya jaatat; eka samanyikaran jeey tyanna parvanagi dete teey samatulya nahee.
udaharane.
vastavik sadish antaral sootra _ 2 madhye {(−1,0,0), (0,1,0), (0,0,1)} haah eka vistar sanch aahe. haah vishisht vistar sanch dekhil eka aadhar aahe. jarr (− 1,0,0) chee jaaga (1,0,0) neey ghetali tarr tee sootra _ 2 chaa vihit aadhar dekhil tayar karel.
tyach jagesathi aanakhi eka vistar sanch {(1,2,3), (0,1,2), (−1,3), (1,1,1)} dware deela jaato, parantu haah sanch aadhar nahee, kaaran too reshiyadrushtya avalambun aahe.
sanch} haah sootra _ 2 chaa vistarlela sanch nahee, kaaran tyacha kalavadhi haah sootra _ 2 madheel serve vectorchi jaaga aahe jyacha shevatcha ghatak shoonya aahe. tee jaaga dekhil {(1,0,0), (0,1,0)} yaa sanchadvare pasarleli aahe, kaaran (1,1,0) heey (1,0,0) aanee (0,1,0) chey resheey sanyojan aahe. asha prakare, pasarleli jaaga sootra 6 nahee, teey shoonyachya barobariche tisare ghatak kadhoon sootra 7 neey olakhale jaau shakate.
rikama sanch haah {(0,0,0)} chaa vistarlela sanch aahe, kaaran rikama sanch haah sootra _ 2 madheel serve sambhavya vector jagancha upasanch aahe aanee {(0,0,0)} haah yaa serve vector jagancha chhedanbindu aahe.
ekmudracha sanch, jithe eka nakaratmak nasalela purnank aahe, too bahupadanchya jaaget pasarato.
pramey.
vyakhyanchi samanata.
vector space over yaa upasanchatil serve resheey sanyojanancha sanch haah samavisht karnyacha sarvat lahan resheey upkshetra aahe.
spanning sanch kiman resheey swatantra sanch aakar aahe.
vector spacechya pratyek spanning sanchamadhye vectorchya konatyahi resheey swatantra sanchaitke kiman anek ghatak asane aavashyak aahe.
spanning seet besisparyant qamee keela jaau shakato.
eka maryadit-aayami vector space asoo dya. aavashyak asalyas vahak takoon, vistarlelya vahakancha konatahi sanch aadharaparyant qamee keela jaau shakato (i.eae. jarr sanchamadhye resheey avalambun asalele vahak asateel). nivadicha svayamsiddha siddhant jarr asel tarr heey maryadit pariman asalelya gruhitkashivay kharey aahe. heey asehi soochit karate qee jevha maryadit-aayami asel tevha aadhar haah kiman vistar sanch asato.
samanyikaran.
antaralatil bindunchya kalavadhichi vyakhya samanya kartana, jarr kramvari sampoorn ground setchya shrenichya barobarichi asel tarr matroidchya ground setchya upasanchala spanning seet mhanatat.
vector spacechi vyakhya modulesmadhye dekhil samanyikrut kelee jaau shakate. eka-module aanee ghatakancha sangrah deela aahe..., chey, upaghatkadvare pasarlele,..., hii chakriy moduleschee berries aahe
sootra _ 32
ghatakanchya serve "aar"-rekheey sanyogancha samavesh aahe. vector spaceschya babatit, 'aaa' chya konatyahi upasanchadvare pasarlele 'aaa' chey upghatak heey tyaa upasanchasah asalelya serve upaghatakanche chhedanbindu aahe.
bund resheey kalavadhi (karyatmak vishleshan).
karyatmak vishleshanamadhye, vectorchya sanchacha bund resheey kalavadhi haah kiman bund sanch asato jyamadhye tyaa sanchacha resheey kalavadhi asato.
samaja tee eka normed vector space aahe aanee tyacha konatahi rikama nasalela upasanch asoo dya. sootra _ 33 kinwa sootra _ 34 dware darshavilela bund resheey kalavadhi, jyaa serve bund resheey upasthanancha samavesh aahe tyanche chhedanbindu aahe.
yache eka ganiteey sootra mhanje
[0,1] yaa madhyantaravaril "xn" yaa karyanchya sanchacha bund resheey kalavadhi, jethe "n" haah eka nakaratmak nasalela purnank aahe, too vaparlelya nikashavar avalambun asato. jarr "l" 2 manak vaparla gela, tarr bund resheey kalavadhi haah madhyantaravaril varg-ekatmik karyanchi hilbert jaaga aahe. parantu jarr kamal manak vaparale gele tarr bund resheey kalavadhi haah madhyantaravaril satatchya karyanchi jaaga asel. donhi babatit, bund resheey kalavadhimadhye ashee karye asatat jee bahupadi nasatat aanee tyamule tee resheey kalavadhimadhyech nasatat. tathapi, bund resheey kalavadhitil karyanchya sanchachi mukhyata hii satatyachi mukhyata aahe, jee bahupadichya sanchaevadhich mukhyata aahe.
nondi.
bund resheey kalavadhimadhye sanchacha resheey kalavadhi daat asato. shivay, khaleel shabdasangrahat namood kelyapramane, bund resheey kalavadhi haah kharokhar resheey kalavadhicha ant aahe.
bund resheey upasthananshi vyavhar kartana bund resheey vistar mahatvache asatat (jeey swatah atyant mahatvache aahet, irizcha lemma pahaa).
eka upayukt shabd.
eka samanya jaaga asoo dya aanee tyacha konatahi rikama nasalela upasanch asoo dya. mug
(eae) </math> heey 'x' chey bund resheey upkshetra aahe jyamadhye 'e',
 . sootra _ 36, mhanje. sootra _ 33 heey sootra _ 38 chey samapan aahe,
 . sootra _ 39