rank Additionally system of equation only independent atfect Gawp (A) possible ( Alb ) easily process form rang A. Proposition with Definition

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1 Defiion nexivnol numer ler dependen rows mrix sid row Gwp elimion mehod does no fec h numer end process i possile esily red rng fc for mrix form der zz rn rnk wih m dcussion i holds rr o Proposiion ler equion x solvle nd only l ddiionlly i holds ho i

2 mrix Formul Proposiion For every Em ] i holds DMENSON FORMUL dim Kern n n under deermed homogeneous ler xo wih E Dum nd mn hs non rivil soluions Pro ler e Ri hs x wih unique soluion squre nd only mrix e w rnk s resul nd nm which c 5 Th Ff n r n Grp r dim Kern Gwp fn elimion mehod we o gives :/! non rivil soluion x : L hs r o nd # F hs soluion h s dzx unique only p hs soluion s n nr rng r given u y? di ER r l nd y previous Proposiion 8 for n r shows h h unique

3 Defiion regulr sid cse Proposiion for vecrs now sid e u e i Orwe holds i mrix mehix squre squre Ri e sgulr sgulr e squre non or mrix Ri E : i equivlen : lerly dependen re rnk Co d w regulr x Ce xe f x solvle hs ER given mrix B look B for n x h y x e given x mrix solves consider show soluiliy exs re R esy very h such verse equivlen such verile nme R e exence kes which B mrix implicion h so One denoed nd Mrix we soluion B h for every rivil only verse soluion unique hs o for every x ssume mrix verse For fee x

4 dep f sep dep rows cully i holds : 1 2 verile regulr regulr column re liver dependen Moreover one cn prove h for numer numer l rows columns every HER grees Ln or words mn wih one cow compue rng y coung eir mximl numer or columns Noe h verile nd verse i holds h : D D w For which sys B e R h lso verile nd Y # regulr we hve CȦBJ s B he B B CB BT f E Ri regulr TTL T foo T i T s

5 h Lorden k h How cn we compue gmp verse mrix? lgorihm Noe h give e R e ; i column Tf we wn fd : we hve priculr ei ei ED Hei ei h i column x solves x ei Emem iii will give sum! We cn solve h wih Gmp lgorihm We now cn do sme for ll or ei sme ime considerg i Tf we conue m f re Gu 1 O l lol group unil we rech we here ho E

6 rm Exmple :L : : :L i : :L : i : i c % i l : Ei :L :c : Deermns Given C i j e 112 we defe di E R s ou / nn?? s sign Chrm nu where Sw se ll nd sign C r E L permuions { 12 g sign heh on pnmu We recll h permuion TE Sw from 112 ] 144 } Such mop mp oed y composg elemenry rnsposiions wo elemens

7 15 More sign rnsposiion 2 sign 2 sign permuion or Go o Zu 13h C Exmple Le u g pnnrwhehiou S3 for snce mop order rnsposiion We cn o y numers 123 wo numers we 2 3 [ 23 ] elemens posiion 2 nd 3 devoe y Zig [ i j ] sy o l 32 rnsposiion posiion i wih h posiion j or [ 23 ] Exmple 2 signs C 1 chngg Ech ime we we perform order chnge rnsposiion generl we elemen h exmple le me 4 o 2 We consider re Oz Eg Ni ru z We hve N 23 ]o[ 4 C r2 ] posiions posiions 2 3 U u 1 2 ] sign or C 1J

8 z l C Check defiion deermn cse h 2 nd he 3 n!: di TE Sz sign on rr rz U possile permuions wo numers re on C on r 4i T on K 4 r 2 z firs cse re 1 2z no rnsposiion sign Co L while second cse re one rnsposiion nd sign C D s resul ou 1 31z n * : : 12 3 sign F! re 1132 signs L r 3 21 C sign 021/3 signer C 1 y y o sign 72+1 r L : signer C 421 r? l di rnz zz zn nz 923 zn n

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