
samarupata virodhi

ganitat, samarupata virodhi heey gunakarasah sanchanvar paribhashit kelele eka prakarche karya aahe jeey gunakar kram ulatate. antiaotomorphism haah eka bijective antihomomorphism aahe, i.eae. eka sanchapasun swatahparyant, eka antiisomorphism. byjectivitivarun asey disun yete qee antiaotomorphism inverse asatat aanee antiaotomorphismcha inverse dekhil antiaotomorphism aahe.
vyakhya.
anaupacharikpane, samarupatavirodhi haah eka nakasha aahe joou gunakar kram badalato. aupcharikpane, sanrachana sootra _ 1 aanee sootra _ 2 madheel samarupata virodhi sootra heey eka samarupata sootra _ 3 aahe, jethe sootra _ 4 heey eka sanch mhanoon sootra _ 2 chya barobariche asate, parantu tyache gunakar sootra _ 2 varr paribhashit kelelya gunakarakade ulatale aahe. sootra _ 2 varr sootra _ 8 dware gunakar darshavitana, sootra _ 10 dware darshavilelya sootra _ 4 vareel gunakar sootra _ 11 dware paribhashit keela jaato. object sootra _ 4 laa sootra _ 2 chya viruddha object mhanatat (anukrame, viruddha gut, viruddha bijganit, viruddha shreni ii.).
hii vyakhya samarupata sootra _ 14 chya samatulya aahe (nakasha lagoo karnyapurvi kinwa nantar operation ulat karane samatulya aahe). aupcharikpane, sootra _ 1 laa sootra _ 16 pathvane aanee nakashanvaril olakh mhanoon kaam karane haah eka karyakarta (kharokhar, eka hastakshep) aahe.
udaharane.
gut siddhantat, samarupatavirodhi haah done gatanmadhil eka nakasha aahe joou gunakar kram ulatato. tarr jarr gut samarupatavirodhi asel, 
sarvansathi 'x', 'x' madheel 'y'.
x" laa "x" −1 pathavnara nakasha heey gut pratiatomorphism udaharan aahe. aanakhi eka mahattvache udaharan mhanje resheey bijganitetil transpose operation, jeey roow vectorla collum vectorvar gheuun jaate. konatehi vector-matricks samikaran samatulya samikaranat rupantarit kele jaau shakate jethe ghatakancha kram ulat keela jaato.
matricks, antiaotomorphism udaharan transpose nakashadvare dile jaate. ulatya aanee transposing yaa donhimule antiaotomorphism midatat, tyanchi rachana automorphism aahe. yaa antarvirodhala anekada virodhabhasi nakasha mhanatat aanee heey samanya resheey gatachya bahya automorphism udaharan dete, jethe 'eff' heey eka kshetra aahe, kevha aanee, kinwa aanee (aay.eae., gatansathi, aanee).
ring siddhantat, antihomomorphism haah done ringjamadhil eka nakasha aahe joou jode jatan karato, parantu gunakar kram ulatato. jarr aanee fakt jarah
sarvansathi 'x', 'x' madheel 'y'.
k" kshetravaril algebrasathi, "φ" haah antarnihit vector jaagecha "k"-rekheey nakasha asane aavashyak aahe. jarr antarnihit kshetramadhye involution asel tarr tyaeivaji khalee sanyugma transpospramane, konee "φ" laa sanyugma-resheey mhanoon vicharoo shakate.
hastakshep.
asey anekada ghadate qee antiaotomorphism heey involution asatat, i.eae. antiaotomorphismcha varg haah olakh nakasha aahe; hyanna ess dekhil mhanatat. udaharanarth, konatyahi gatamadhye, jyaa nakashavarun 'x' tyachya viruddha 'x'-1 varr pathavala jaato, too eka anaichhik pratiaatmarupata aahe.
anaichhik antiaotomorphism asalelya ringla *-ring mhanatat aanee heey udaharanancha eka mahattvacha varg tayar karatat.
malmatta.
jarr srot "x" kinwa lakshya "y" heey parivartaneey asel, tarr samarupata virodhi hii samarupata sarkhich gosht aahe.
done samarupata virodhi rachananchi rachana nehmich samarupata asate, kaaran donada kram ulat kelyane kram sanrakshit rahato. samarupatesah samarupateviruddhachya rachnemule aanakhi eka samarupateviruddha rachana milate.