W. B. Vasantha Kandasamy Florentin Smarandache NEUTROSOPHIC BILINEAR ALGEBRAS AND THEIR GENERALIZATIONS

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2 W. B. Vsnh Kndsmy Florenn Smrndche NEUTROSOPHIC BILINEAR ALGEBRAS AND THEIR GENERALIZATIONS Svensk fyskrkve Sockholm, Sweden 00

3 Svensk fyskrkve (Swedsh physcs rchve) s publsher regsered wh he Royl Nonl Lbrry of Sweden (Kunglg bbloeke), Sockholm. Posl ddress for correspondence: Näsbydlsvägen 4/, 833 Täby, Sweden Peer-Revewers: Prof. Mhàly Bencze, Deprmen of Mhemcs Áprly Ljos College, Brşov, Romn Prof. Ion Prscu, Deprmen of Mhemcs, Fr, Buzes College Crov, Romn Dr. Fu Yuhu, 3-603, Lufngbel Lufng Sree, Choyng dsrc, Bejng, 0008 P. R. Chn Copyrgh Auhors, 00 Copyrgh Publcon by Svensk fyskrkve, 00 Copyrgh noe: Elecronc copyng, prn copyng nd dsrbuon of hs book for non-commercl, cdemc or ndvdul use cn be mde by ny user whou permsson or chrge. Any pr of hs book beng ced or used howsoever n oher publcons mus cknowledge hs publcon. No pr of hs book my be reproduced n ny form whsoever (ncludng sorge n ny med) for commercl use whou he pror permsson of he copyrgh holder. Requess for permsson o reproduce ny pr of hs book for commercl use mus be ddressed o he Auhor. The Auhor rens hs rghs o use hs book s whole or ny pr of n ny oher publcons nd n ny wy he sees f. Ths Copyrgh Agreemen shll remn vld even f he Auhor rnsfers copyrgh of he book o noher pry. ISBN:

4 CONTENTS Dedcon 5 Prefce 7 Chper One INTRODUCTION TO BASIC CONCEPTS 9. Inroducon o Blner Algebrs nd her Generlzons 9. Inroducon o Neurosophc Algebrc Srucures Chper Two NEUTROSOPHIC LINEAR ALGEBRA 5. Neurosophc Bvecor Spces 5. Srong Neurosophc Bvecor Spces 3.3 Neurosophc Bvecor Spces of Type II 50.4 Neurosophc Bnner Produc Bvecor Spce 98 3

5 Chper Three NEUTROSOPHIC n-vector SPACES Neurosophc n-vecor Spce Neurosophc Srong n-vecor Spces Neurosophc n-vecor Spces of Type II 307 Chper Four APPLICATIONS OF NEUTROSOPHIC n-linear ALGEBRAS 359 Chper Fve SUGGESTED PROBLEMS 36 FURTHER READING 39 INDEX 396 ABOUT THE AUTHORS 40 4

6 DEDICATION Ths book s dedced o he memory of Thvhru Kundrkud Adglr ( July 95-5 Aprl 995), sprul leder who worked relessly for socl 5

7 developmen nd communl hrmony. Hs powerful wrngs nd speeches provded mssve mpeus o he promoon of Tml lerure nd culure. Movng beyond he relm of relgon, he ws lso cvely nvolved n brngng chnge he grssroos level. Durng hs lfeme nd fer, he ws celebred for hs successful effors o develop povery-srcken vllges round Kundrkud hrough plnnng, suppor nd connued nervenon. Hs nves for rurl developmen erned prse even from Indr Gndh, he hen Prme Mnser of Ind. Alhough he heded fmed relgous cener, Kundrkud Adglr mnned scenfc oulook owrds he world nd pd specl enon o he educonl uplf of poor people cross cse, communy or relgon. The servces rendered by hm o socey re endless, nd he dedcon of hs book s smll gesure o py homge o h gre mn. 6

8 PREFACE Ths book nroduces he concep of neurosophc blner lgebrs nd her generlzons o n-lner lgebrs, n>. Ths book hs fve chpers. The reder should be well-versed wh he noons of lner lgebrs s well s he conceps of blner lgebrs nd n- lner lgebrs. Furher he reder s expeced o know bou neurosophc lgebrc srucures s we hve no gven ny deled lerure bou. The frs chper s nroducory n nure nd gves few essenl defnons nd references for he reder o mke use of he lerure n cse he reder s no horough wh he bscs. The second chper dels wh dfferen ypes of neurosophc blner lgebrs nd bvecor spces nd proves severl resuls nlogous o lner blgebr. In chper hree he uhors nroduce he noon of n-lner lgebrs nd prove severl heorems reled o hem. Mny of he clsscl heorems for neurosophc lgebrs re proved wh ppropre modfcons. Chper four ndces he probble pplcons of hese lgebrc srucures. The fnl chper suggess bou 80 nnovve problems for he reder o solve. 7

9 The neresng feure of hs book s h hs over 5 llusrve exmples, hs s mnly provded o mke he reder undersnd hese new conceps. Ths book conns over 60 heorems nd hs nroduced over 00 new conceps. The uhors deeply cknowledge Dr. Kndsmy for he proof redng nd Meen nd Km for he formng nd desgnng of he book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 8

10 Chper One INTRODUCTION TO BASIC CONCEPTS Ths chper hs wo secons. In secon one bsc noons bou blner lgebrs nd n-lner lgebrs re reclled. In secon wo n nroducon o ndeermncy nd lgebrc neurosophc srucures essenl for hs book re gven.. Inroducon o Blner Algebrs nd her Generlzons In hs secon we jus recll some necessry defnons bou blner lgebrs. DEFINITION..: Le (G, +, ) be bgroup where G = G G ; bgroup G s sd o be commuve f boh (G, +) nd (G, ) re commuve. DEFINITION..: Le V = V V where V nd V re wo proper subses of V nd V nd V re vecor spces over he sme feld F h s V s bgroup, hen we sy V s bvecor spce over he feld F. If one of V or V s of nfne dmenson hen so s V. If V nd V re of fne dmenson so s V; o be more precse f V s 9

11 of dmenson n nd V s of dmenson m hen we defne dmenson of he bvecor spce V = V V o be of dmenson m + n. Thus here exss only m + n elemens whch re lnerly ndependen nd hs he cpcy o genere V = V V. The mporn fc s h sme dmensonl bvecor spces re n generl no somorphc. Exmple..: Le V = V V where V nd V re vecor spces of dmenson 4 nd 5 respecvely defned over ronls where V = {( j ) j Q}, collecon of ll mrces wh enres from Q. V = {Polynomls of degree less hn or equl o 4 wh coeffcens from Q}. Clerly V s fne dmensonl bvecor spce over Q of dmenson 9. In order o vod confuson we cn follow he followng convenon whenever essenl. If v V = V V hen v V or v V f v V hen v hs represenon of he form (x, x, x 3, x 4, 0, 0, 0, 0, 0) where (x, x, x 3, x 4 ) V f v V hen v = (0, 0, 0, 0, y, y, y 3, y 4, y 5 ) where (y, y, y 3, y 4, y 5 ) V. DEFINITION..3: Le V = V V be bgroup. If V nd V re lner lgebrs over he sme feld F hen we sy V s lner blgebr over he feld F. If boh V nd V re of nfne dmensonl lner lgebrs over F hen we sy V s n nfne dmensonl lner blgebr over F. Even f one of V or V s nfne dmenson hen we sy V s n nfne dmensonl lner blgebr. If boh V nd V re fne dmensonl lner lgebr over F hen we sy V = V V s fne bdmensonl lner blgebr. Exmples..: Le V = V V where V = {se of ll n n mrces wh enres from Q} nd V be he polynoml rng Q[x]. V = V V s lner blgebr over Q nd he lner blgebr s n nfne dmensonl lner blgebr. Exmple..3: Le V = V V where V = Q Q Q beln group under +, V = {se of ll 3 3 mrces wh enres from Q} hen V = V V s bgroup. Clerly V s lner 0

12 blgebr over Q. Furher dmenson of V s ; V s dmensonl lner blgebr over Q. The sndrd bss s {(0 0), ( 0 0), (0 0 )} , 0 0 0, 0 0 0, 0 0, 0 0, , 0 0 0, 0 0 0, DEFINITION..4: Le V = V V be bgroup. Suppose V s lner blgebr over F. A non empy proper subse W of V s sd o be lner subblgebr of V over F f. W = W W s subbgroup of V = V V.. W s lner sublgebr over F.. W s lner sublgebr over F. For more refer [48, 5-]. For n-lner lgebr of ype I nd II, refer[54-5].. Inroducon o Neurosophc Algebrc Srucures In hs secon we jus recll some bsc neurosophc lgebrc srucures essenl o mke hs book self conned one. For more refer [36-43, 53]. In hs secon we ssume felds o be of ny desred chrcersc nd vecor spces re ken over ny feld. We denoe he ndeermncy by I, s wll mke confuson, s denoes he mgnry vlue, vz. = h s =. The ndeermncy I s such h I. I = I = I. Here we recll he noon of neurosophc groups. Neurosophc groups n generl do no hve group srucure.

13 DEFINITION..: Le (G, *) be ny group, he neurosophc group s genered by I nd G under * denoed by N(G) = { G I, *}. Exmple..: Le Z 7 = {0,,,, 6} be group under ddon modulo 7. N(G) = { Z 7 I, + modulo 7} s neurosophc group whch s n fc group. For N(G) = { + bi /, b Z 7 } s group under + modulo 7. Thus hs neurosophc group s lso group. Exmple..: Consder he se G = Z 5 \ {0}, G s group under mulplcon modulo 5. N(G) = { G I, under he bnry operon, mulplcon modulo 5}. N(G) s clled he neurosophc group genered by G I. Clerly N(G) s no group, for I = I nd I s no he deny bu only n ndeermne, bu N(G) s defned s he neurosophc group. Thus bsed on hs we hve he followng heorem: THEOREM..: Le (G, *) be group, N(G) = { G I, *} be he neurosophc group.. N(G) n generl s no group.. N(G) lwys conns group. Proof: To prove N(G) n generl s no group s suffcen we gve n exmple; consder Z 5 \ {0} I = G = {,, 4, 3, I, I, 4 I, 3 I}; G s no group under mulplcon modulo 5. In fc {,, 3, 4} s group under mulplcon modulo 5.N(G) he neurosophc group wll lwys conn group becuse we genere he neurosophc group N(G) usng he group G nd I. So G N(G); hence N(G) wll lwys conn group. Now we proceed ono defne he noon of neurosophc subgroup of neurosophc group. DEFINITION..: Le N(G) = G I be neurosophc group genered by G nd I. A proper subse P(G) s sd o be neurosophc subgroup f P(G) s neurosophc group.e. P(G) mus conn (sub) group of G.

14 Exmple..3: Le N(Z ) = Z I be neurosophc group under ddon. N(Z ) = {0,, I, + I}. Now we see {0, I} s group under + n fc neurosophc group {0, + I} s group under + bu we cll {0, I} or {0, + I} only s pseudo neurosophc groups for hey do no hve proper subse whch s group. So {0, I} nd {0, + I} wll be only clled s pseudo neurosophc groups (subgroups). We cn hus defne pseudo neurosophc group s neurosophc group, whch does no conn proper subse whch s group. Pseudo neurosophc subgroups cn be found s subsrucure of neurosophc groups. Thus pseudo neurosophc group hough hs group srucure s no neurosophc group nd neurosophc group cnno be pseudo neurosophc group. Boh he conceps re dfferen. Now we see neurosophc group cn hve subsrucures whch re pseudo neurosophc groups whch s evden from he followng exmple. Exmple..4: Le N(Z 4 ) = Z 4 I be neurosophc group under ddon modulo 4. Z 4 I = {0,,, 3, I, + I, I, 3I, + I, + 3I, + I, + I, + 3I, 3 + I, 3 + I, 3 + 3I}. o( Z 4 I ) = 4. Thus neurosophc group hs boh neurosophc subgroups nd pseudo neurosophc subgroups. For T = {0,, + I, I} s neurosophc subgroup s {0 } s subgroup of Z 4 under ddon modulo 4. P = {0, I} s pseudo neurosophc group under + modulo 4. DEFINITION..3: Le K be he feld of rels. We cll he feld genered by K I o be he neurosophc feld for nvolves he ndeermncy fcor n. We defne I = I, I + I = I.e., I + + I = ni, nd f k K hen k.i = ki, 0I = 0. We denoe he neurosophc feld by K(I) whch s genered by K I h s K(I) = K I. K I denoes he feld genered by K nd I. Exmple..5: Le R be he feld of rels. The neurosophc feld of rels s genered by R nd I denoed by R I.e. R(I) clerly R R I. 3

15 Exmple..6: Le Q be he feld of ronls. The neurosophc feld of ronls s genered by Q nd I denoed by Q(I). DEFINITION..4: Le K(I) be neurosophc feld we sy K(I) s prme neurosophc feld f K(I) hs no proper subfeld, whch s neurosophc feld. Exmple..7: Q(I) s prme neurosophc feld where s R(I) s no prme neurosophc feld for Q(I) R(I). DEFINITION..5: Le K(I) be neurosophc feld, P K(I) s neurosophc subfeld of P f P self s neurosophc feld. K(I) wll lso be clled s he exenson neurosophc feld of he neurosophc feld P. We cn lso defne neurosophc felds of prme chrcersc p (p s prme). DEFINITION..6: Le Z p = {0,,,, p } be he prme feld of chrcersc p. Z p I s defned o be he neurosophc feld of chrcersc p. Infc Z p I s genered by Z p nd I nd Z p I s prme neurosophc feld of chrcersc p. Exmple..8: Z 7 = {0,,, 3,, 6} be he prme feld of chrcersc 7. Z 7 I = {0,,,, 6, I, I,, 6I, + I, + I,, 6 + 6I } s he prme feld of chrcersc 7. DEFINITION..7: Le G(I) by n ddve beln neurosophc group nd K ny feld. If G(I) s vecor spce over K hen we cll G(I) neurosophc vecor spce over K. Elemens of hese neurosophc felds wll lso be known s neurosophc numbers. For more bou neurosophy plese refer [36-43]. We see Z n I = {I Z n } s neurosophc feld clled pure neurosophc feld. Lkewse QI, RI nd Z p I re neurosophc felds where p s prme. Thus Z 5 I = {0, I, I, 3I, 4I} s pure neurosophc feld. For more bou neurosophc vecor spces plese refer [53]. 4

16 Chper Two NEUTROSOPHIC LINEAR BIALGEBRA In hs chper we nroduce he noon of neurosophc lner blgebrs nd descrbe few properes bou hem. Srong neurosophc lner blgebr re lso nroduced. Ths chper hs four secons. In secon one, we nroduce he new noon of neurosophc bvecor spce. Srong neurosophc bvecor spces re nroduced n secon wo. Secon hree nroduces he noon of neurosophc bvecor spce of ype III. Secon four sudes he bnner produc n srong neurosophc bvecor spce.. Neurosophc Bvecor Spces In hs secon we nroduce he noon of neurosophc bvecor spces nd sudy her properes. 5

17 DEFINITION..: Le V = V V where ech V s neurosophc vecor spce over he sme feld F nd V V j, V V j nd V j V ;, j, hen we defne V o be neurosophc bvecor spce over he rel feld F. Noe: We ssume here F s jus rel feld h s F s Q or Z n or R or C. (n prme n < ). We wll llusre hs by some smple exmples. Exmple..: Le V = Q I = N(Q) = { + bi, b Q} be neurosophc vecor spce over Q. Tke V = c b d,b,c,d N(Q), neurosophc vecor spce over Q. V = V V s neurosophc bvecor spce over Q. Exmple..: Le V = V V = {N(Q)[x]} {(, b, c), b, c N(Q)}. V s neurosophc bvecor spce over Q. Now we wll defne qus neurosophc bvecor spce. DEFINITION..: Le V = V V be such h V s vecor spce over he rel feld F nd V s neurosophc vecor spce over F. We defne V =V V o be qus- neurosophc bvecor spce over F. We wll gve some exmples of qus neurosophc bvecor spces. Exmple..3: Le V = V V where V = {Z 7 [x] ll polynomls n he vrble x wh coeffcens from Z 7 } s vecor spce over Z 7 nd V = {(Z 7 I Z 7 I Z 7 I Z 7 I) = {(, b, c, d), b, c, d Z 7 I}} s neurosophc vecor spce over 6

18 Z 7. Then V = V V s qus neurosophc bvecor spce over Z 7. Exmple..4: Le V = V V where V = {Q Q Q Q R} = {(, b, c, d, e), b, c, d Q nd e R} s vecor spce over Q nd V = {QI Q QI Q QI} = {(, b, c, d, e), c, e QI nd b, d Q} s neurosophc vecor spce over Q. Thus V = V V s qus neurosophc bvecor spce over Q. Now we defne subsrucures of hese srucures. DEFINITION..3: Le V = V V be neurosophc bvecor spce over he feld F. Le W = W W V V be such h W s neurosophc bvecor spce over F, hen we defne W o be neurosophc bvecor subspce of V over F. We wll llusre hs by some exmples. Exmple..5: Le V = V V b = {Z 3 I Z 3 I Z 3 I Z 3 I},b,c,d N(Z 3 ) c d be neurosophc bvecor spce over he feld Z 3. Le W = W W b = {(, b, 0, 0), b Z 3 I},b,c,d Z 3 I c d V V ; W s neurosophc bvecor subspce of V over he feld Z 3. Exmple..6: Le V = V V = N (Q) [x] b e g c d f h,b,c,d,e,f,g,h QI 7

19 be neurosophc bvecor spce over he feld Q. Le W = W W = QI [x] b e g,b,e,g QI V V, W s neurosophc bvecor subspce of V over he feld Q. DEFINITION..4: Le V = V V be neurosophc bvecor spce over he feld F. Le W = W W V V be such h W s only qus neurosophc bvecor spce over F; h s one of W or W s only neurosophc vecor spce over F nd oher s jus vecor spce over he feld F; hen we cll W o be pseudo qus neurosophc bvecor subspce of V over he feld F. Exmple..7: Le V = V V where V = (Z 5 I Z 5 I Z 5 I) neurosophc vecor spce over Z 5 nd V = N(Z 5 )[x] neurosophc bvecor spce over Z 5. V = V V s neurosophc bvecor spce over he feld Z 5. Tke W = W W = {Z 5 I {0} {0}} {Z 5 I [x]} V V ; W s pseudo qus neurosophc bvecor subspce of V over Z 5. Exmple..8: Le V = V V = b c d e f,b,c,d,e,f,g,h, RI g h b b b3 b4 b5,b N(Q); 5 8

20 be neurosophc bvecor spce over he feld Q. Tke W = W W = b c 0 d e,b,c,d,e,f QI 0 0 f Q; V V ; W s pseudo qus neurosophc bvecor subspce of V over he feld Q. DEFINITION..5: Le V = V V be neurosophc bvecor spce over he feld F. Suppose W = W W V V s such h W s jus bvecor spce over he feld F hen we defne W o be pseudo bvecor subspce of V over he feld F. We wll gve some exmples of hs noon. Exmple..9: Le V = V V = b,b,c,d N(Z ) c d ({N(Z ) N(Z ) N(Z ) N(Z )) be neurosophc bvecor spce over he feld Z. Le W = W W = b,b,c,d Z c d {(, b, c, d), b, c, d Z } V V be bvecor spce over Z. Thus W s pseudo bvecor subspce of V over he feld Z. 9

21 Exmple..0: Le V = V V = N(Q); 6 {N(Q) N(Q) N(Q) N(Q)} be neurosophc bvecor spces over he feld Q. Tke W = W W = Q; 6 (Q Q {0} {0})} V V ; W s pseudo bvecor subspce of V over he feld Q. DEFINITION..6: Le V = V V be neurosophc bvecor spce over he feld F. Le W = W W V V be such h W s neurosophc bvecor spce over he subfeld K F. Then we cll W o be neurosophc specl bvecor subspce of V over he subfeld K of F. We wll gve some exmples. Exmple..: Le V = V V = {RI[x]} b,b,c,d RI c d be neurosophc bvecor spce over he feld R. Tke W = W W = {QI [x]} b,b,d RI 0 d 0

22 V V ; W s neurosophc specl bvecor subspce of V over he subfeld Q of R. Exmple..: Le V = V V = (RI RI RI) RI; 8 be neurosophc bvecor spce over he feld Q ( = W W = {(QI QI QI)} ). Tke W RI; 4 V V, W s neurosophc specl bvecor subspce of V over he subfeld Q Q ( ). DEFINITION..7: Le V = V V be neurosophc bvecor spce over he feld F. If V hs no neurosophc specl bvecor subspce hen we cll V o be neurosophc specl smple bvecor spce over F. Exmple..3: Le V = V V = {QI QI} {QI[x]} be neurosophc bvecor spce over he feld F = Q. V s neurosophc specl smple bvecor spce over Q s Q hs no proper subfeld. Exmple..4: Le V = V V = Z 3 I Z 3 I Z 3 I Z 3 I} b,b,c,d Z 3 I c d be neurosophc bvecor spce over he feld Z 3. V s neurosophc specl smple bvecor spce over Z 3 s Z 3 s prme feld.

23 In vew of hese exmples we hve he followng neresng heorem. THEOREM..: Le V = V V be neurosophc bvecor spce over he feld F. If F s prme feld of chrcersc zero or prme p hen V s neurosophc specl smple bvecor spce over F. Proof: Gven F s prme feld of chrcersc zero or prme p, so F hs no subfelds. Thus for no W = W W V V cn be neurosophc specl bvecor subspce of V = V V s F hs no subfeld. Hence he clm. Now we proceed ono defne he noon of neurosophc blner lgebr. DEFINITION..8: Le V = V V where boh V nd V re neurosophc lner lgebrs over he feld F, hen we defne V o be neurosophc blner lgebr over F. We wll llusre hs by some smple exmples. Exmple..5: Le V = V V = b,b,c,d QI c d (QI QI QI QI QI)} be neurosophc blner lgebr over Q. Exmple..6: Le V = V V = 0,b,c Z 9 I b c {Z 9 I [x]}, V s neurosophc blner lgebr over he feld Z 9.

24 Now s n cse of neurosophc bvecor spces we cn defne he followng subsrucures. DEFINITION..9: Le V = V V be neurosophc blner lgebr over he feld F. If W = W W V V s neurosophc blner lgebr over he feld F hen we cll W o be neurosophc blner sublgebr of V over he feld F. We gve n exmple. Exmple..7: Le V = V V = b c 0 d e,b,c,d,e,f QI 0 0 f {QI QI QI QI} be neurosophc blner lgebr over he feld Q. Choose 0 0 W = 0 b 0,b,c QI 0 0 c {QI {0} {0} QI} = W W V V ; W s neurosophc blner sublgebr of V over Q. DEFINITION..0: Le V = V V be neurosophc blner lgebr over he feld F. Le W = W W V V be neurosophc blner lgebr over subfeld K F; hen we defne W o be neurosophc specl blner sublgebr of V over he subfeld K of F. If V hs no specl neurosophc blner sublgebr s hen we cll V o be specl neurosophc smple blner lgebr or neurosophc specl smple blner lgebr. 3

25 We wll llusre hs by some smple exmples. Exmple..8: Le V = V V = b,b,c,d RI c d {RI[x] be neurosophc blner lgebr over he feld R. Tke W = W W = b,b,c,d QI c d {QI[x]} V V ; W s neurosophc specl blner lgebr over he subfeld Q of he feld R. Exmple..9: Le V = V V 3 = QI; {RI [x]} be neurosophc blner lgebr over Q. Clerly V s neurosophc specl smple blner lgebr. Exmple..0: Le V = V V = {Z 7 I [x]} {Z 7 I Z 7 I Z 7 I} be neurosophc blner lgebr over he feld Z 7. V s neurosophc smple blner lgebr. In vew of hese exmples we hve he followng heorem, he proof of whch s lef s n exercse for he reder. THEOREM..: Le V = V V be neurosophc blner lgebr over he feld F, where F s prme feld (.e., F hs no subfelds oher hn self). V s neurosophc specl smple blner lgebr. 4

26 DEFINITION..: Le V = V V be neurosophc blner lgebr over feld F. Suppose W = W W V V nd f W s only blner lgebr over he feld F, hen we cll W o be pseudo blner sublgebr of V over he feld F. Exmple..: Le V = V V = b,b,c,d N(Q) c d {Q QI QI Q} be neurosophc blner lgebr over Q, where W = W W = b,b,c,d Q c d {Q {0} {0} {0}} V V ; W s pseudo blner sublgebr of V over he feld F. Exmple..: Le V = V V = b c d e f,b,c,d,e,f,g,h,k N(R) g h k {(, b, c, d), b, c, d N(Q)} be neurosophc blner lgebr over he feld Q. Tke W = W W = b c 0 d e,b,c,d,e,f R 0 0 f 5

27 {(, b, 0, d), b, d Q} V V ; W s pseudo blner sublgebr of V over Q. DEFINITION..: Le V = V V be neurosophc blner lgebr over he feld F. Le W = W W V V be proper bsubse of V whch s jus neurosophc bvecor spce over he feld F. We defne W o be pseudo neurosophc bvecor subspce of V over F. We wll llusre hs by some smple exmples. Exmple..3: Le V = V V = n x QI,= 0,,,,n = 0 b,b,c,d QI c d be neurosophc blner lgebr over he feld Q. Le W = W W = 5 x QI,= 0,,,,5 = 0 0 b 0,b QI V V, W s pseudo neurosophc bvecor subspce of V over Q. Exmple..4: Le V = V V = b c d e f,b,c,d,e,f,g,h, ZI g h 6

28 n = 0 x ZI, = 0,,,...,n;n be neurosophc blner lgebr over he feld Z. Le W = W W = b c d 0 0,b,c,d ZI = x 0 ZI, 6 V V, W s only pseudo neurosophc bvecor subspce of V s we see b c b c d 0 0 d = d bd cd bd b c W. Smlrly f we ke = x 6 + 3x + nd b = (Ix 6 + 3Ix + I) (3x 4 + x + I) = 6Ix 0 + 4Ix 8 + Ix 6 + 9Ix 5 + 6Ix 3 + 3Ix + 3Ix 4 + Ix + I W bu, b W. Thus W s only neurosophc bvecor subspce of V nd no neurosophc blner sublgebr of V over Z. We hve he followng neresng heorem, he proof of whch s lef s n exercse for he reder. THEOREM..3: Le V = V V be neurosophc blner lgebr over he feld F. V s clerly neurosophc bvecor spce over he feld F. If V s neurosophc bvecor spce over he feld F hen n generl V s no neurosophc blner lgebr over he feld F. DEFINITION..3: Le V = V V where V s only neurosophc lner lgebr over he feld F nd V s jus lner lgebr over F hen we defne V = V V o be qus neurosophc blner lgebr over he feld F. 7

29 We llusre hs by some exmples. Exmple..5: Le V = V V = b,b,c,d R c d {( b c d e f), b, c, d, e, f QI} be qus neurosophc blner lgebr over he feld Q. Exmple..6: Le V = V V = b,b,c,d QI c d n x Q ; = 0 V s qus neurosophc blner lgebr over he feld Q. DEFINITION..4: Le V = V V where V s neurosophc vecor spce over he feld F nd V s neurosophc lner lgebr over he feld F. V = V V s defned o be pseudo neurosophc qus blner lgebr over F. We wll llusre hs by some smple exmples. Exmple..7: Le V = V V = b e c d f,b,c,d,e,f QI b,b,c,d N(Q). c d V s pseudo neurosophc qus blner lgebr over he feld Q. 8

30 Exmple..8: Le V = V V = 8 x QI;0 8 = 0 b c d e f,b,c,d,e,f,g,h, RI ; g h V s pseudo neurosophc qus blner lgebr over he feld Q. We cn hve for ny neurosophc blner lgebr V subsrucure whch s pseudo neurosophc qus blner sublgebr of V. DEFINITION..5: Le V = V V be neurosophc blner lgebr over feld F. Suppose W = W W V V such h W s neurosophc vecor spce over F nd W s neurosophc lner lgebr over F hen we defne W o be pseudo neurosophc qus blner sublgebr of V over he feld F. We wll llusre hs by some smple exmples. Exmple..9: Le V = V V = b,b,c,d QI c d {(, b, c, d, e, f), b, c, d, e, f QI} be neurosophc blner lgebr over he feld Q. Tke W = W W = 0,b QI {( 0 c 0 e 0), c, e QI} b 0 9

31 V V W s pseudo neurosophc qus blner sublgebr of V over he feld Q. Exmple..30: Le V = V V = n x Z7I;0 n = 0 b c d e f,b,c,d,e,f,g,h, Z7I g h be neurosophc blner lgebr over he feld Z 7. Choose W = W W = n x Z7I;0 n = b 0 0,b,c,e Z7I c e 0 V V ; W s only pseudo neurosophc qus blner sublgebr of V over he feld Z 7. We see 0 0 b 0 0 W c e 0 Bu b 0 0 b 0 0 W. c e 0 c e 0 30

32 Now we proceed ono defne lner brnsformon of neurosophc bvecor spce nd neurosophc blner lgebr over he feld F. DEFINITION..6: Le V = V V be neurosophc bvecor spce over feld F nd W = W W be noher neurosophc bvecor spce over he sme feld F. Defne T = T T : V = V V W = W W s follows T : V W, =, s jus neurosophc lner rnsformon from V o W. Ths T = T T s neurosophc lner brnsformon of V no W. If W = V hen we cll he neurosophc lner brnsformon s neurosophc lner boperor. We denoe by BN F (V,W) = {se of ll neurosophc lner brnsformons of V = V V o W = W W }; BN F (V,W) = BN F (V, W ) BN F (V, W ). BN F (V,V) = {se of ll neurosophc lner boperors of V o V nd BN F (V, V ) BN F (V, V ) = BN F (V,V). Ineresed reder cn sudy he lgebrc srucures of BN F (V,W) nd BN F (V,V). However we gve n exmple of ech. Exmple..3: Le V = V V = b,b,c QI c 0 b c,b,c,d,e,f QI d e f nd W = W W = {QI QI QI} 5 x QI; 5 = 0 be neurosophc bvecor spces over he feld Q. Defne T = T T : V = V V W = W W where T : V W nd T : V W s follows: 3

33 nd T T b = (, b, c) c 0 b c d e f = { + bx + cx + dx 3 + ex 4 + fx 5, b, c, d, e, f QI}. Clerly T = T T s neurosophc lner brnsformon of V o W. Exmple..3: Le V = V V = b,b,c,d Z 7 I c d {Z 7 I Z 7 I Z 7 I Z 7 I} be neurosophc bvecor spce over he feld Z 7. T = T T : V = V V V = V V, where T : V V nd T : V V s s follows. nd T c b c d = d b T (, b, c, d) = (, b + c, d, + d). I s esly verfed. T s neurosophc lner boperor on V.. Srong Neurosophc Bvecor Spces In hs secon we for he frs me nroduce he noon of srong neurosophc bvecor spces nd sudy hem. 3

34 DEFINITION..: Le V = V V where V nd V re neurosophc ddve beln groups. Suppose V = V V s neurosophc bvecor spce over neurosophc feld F hen we cll V o be srong neurosophc bvecor spce. We wll gve some exmples. Exmple..: Le V = V V = {Z 5 I [x]} {Z 5 I Z 5 I Z 5 I} be srong neurosophc bvecor spce over he neurosophc feld Z 5 I. Exmple..: Le V = V V = b,b,c,d RI c d b c d e f g h j k l,b,c,d,e,f,g,h,, j,k,l QI be srong neurosophc bvecor spce over he neurosophc feld QI. DEFINITION..: Le V = V V be srong neurosophc bvecor spce over he neurosophc feld K. If W = W W V V ; nd f W s srong neurosophc bvecor spce over he neurosophc feld K, hen we cll W o be srong neurosophc bvecor subspce of V over he neurosophc feld F. We wll llusre hs by some smple exmples. Exmple..3: Le V = V V = QI; 6 33

35 QI; 8 be srong neurosophc bvecor spce over he neurosophc feld QI. Tke W = W W 0 3 = 0 4 0,, 3 4 QI 0 0,,, 5 6 QI V V ; W s srong neurosophc bvecor subspce of V over he feld QI. Exmple..4: Le V = V V = {QI QI QI} {QI [x]} be srong neurosophc bvecor spce over he neurosophc feld QI. Tke W = W W = {(QI {0} QI} 8 x QI;0 8 = 0 V V, W s srong neurosophc bvecor subspce of V over QI. Le us defne he noon of srong neurosophc blner lgebr. DEFINITION..3: Le V = V V be neurosophc blner lgebr over he neurosophc feld K, we defne V o be srong neurosophc blner lgebr over K. We wll llusre hs by exmples. 34

36 Exmple..5: Le V = V V = {QI [x]} b,b,d QI 0 d be he srong neurosophc blner lgebr over he neurosophc feld QI. Exmple..6: Le V = V V = {QI QI QI QI QI} b,b,c,d N(Q) c d be srong neurosophc blner lgebr over he neurosophc feld QI. Exmple..7: Le V = V V = {Z I Z I Z I} b c d e f,b,c,d,e,f,g,h, N(Z ) g h be srong neurosophc blner lgebr over he neurosophc feld Z I. DEFINITION..4: Le V = V V be srong neurosophc blner lgebr over he neurosophc feld K. If W = W W V V be srong neurosophc blner lgebr over K hen we defne W o be srong neurosophc blner sublgebr of V over he neurosophc feld K. Exmple..8: Le V = V V = b,b,c,d QI c d 35

37 {(, b, c, d, e, f), b, c, d, e, f QI} be srong neurosophc blner lgebr over he neurosophc feld K = QI. Tke W = W W = 0,d QI 0 d {(, 0, 0, d, 0, f), d, f QI} V V ; W s srong neurosophc blner sublgebr of V over K = QI. Exmple..9: Le V = V V = {Z 7 I [x]} b,b,c,d Z 7 I c d be srong neurosophc blner lgebr over he neurosophc feld Z 7 I. Tke W = W W = n x Z7I; = 0,,,..., 7 = 0 Z I V V ; W s srong neurosophc blner sublgebr of V over he neurosophc feld Z 7 I. DEFINITION..5: Le V = V V be srong neurosophc blner lgebr over he neurosophc feld K. Le W = W W V V where W s jus neurosophc vecor spce over he neurosophc feld K nd W s neurosophc lner sublgebr over he neurosophc feld K. W = W W s defned o be pseudo srong neurosophc lner sublgebr of V over K. We wll llusre hs suon by some exmples. 36

38 Exmple..0: Le V = V V = Z7I; {Z 7 I [x]} be neurosophc blner lgebr over he neurosophc feld Z 7 I. Tke W = W W = b,b,c Z7I 0 c 0 0 ; = 0,,,..., n x n = 0 V V ; W s pseudo srong neurosophc blner sublgebr of V over he feld Z 7 I. Exmple..: Le V = V V = b,b,c,d Z 3 I c d {Z 3 I Z 3 I Z 3 I Z 3 I} be srong neurosophc blner lgebr over he neurosophc feld Z 3 I. Tke W = W W = 0 d,d Z 3 I 0 {(0 0 0 b), b Z 3 I} V V, W s pseudo srong neurosophc blner sublgebr of V over Z 3 I. 37

39 Exmple..: Le V = V V = {QI [x]} b c d e f,b,c,d,e,f,g,h, QI g h be srong neurosophc blner lgebr over he neurosophc feld QI. Tke W = W W = n x QI = b 0,b,d QI 0 0 d V V ; W s pseudo srong neurosophc blner sublgebr of V over he feld QI. DEFINITION..6: Le V = V V be srong neurosophc blner lgebr over he neurosophc feld K. Le W = W W V V be srong neurosophc bvecor spce over he neurosophc feld K. W s defned s he srong pseudo neurosophc bvecor subspce of V over he feld K. We llusre hs by some exmples. Exmple..3: Le V = V V = b c d e f,b,c,d,e,f,g,h, Z3I g h = 0 x Z3I; =,,..., 38

40 be srong neurosophc blner lgebr over he neurosophc feld Z 3 I. Tke W = W W = b c d e f,b,c,d,e,f Z3I = 0 x Z3I;0 9 V V, W s srong pseudo neurosophc bvecor subspce of V over he feld Z 3 I. Exmple..4: Le V = V V = Z I; 6 = 0 x ZI; =,,..., be srong neurosophc blner lgebr over he neurosophc feld Z I. Tke W = W W = ZI;

41 6 = 0 x ZI; =,,...,6 V V, W s pseudo srong neurosophc bvecor subspce of V over Z I. DEFINITION..7: Le V = V V be srong neurosophc blner lgebr over he neurosophc feld K. Tke W = W W V V nd F K (F feld nd s no neurosophc subfeld of K). If W s neurosophc blner lgebr over he feld F hen we defne W o be pseudo srong neurosophc blner sublgebr of V over he subfeld F of he neurosophc feld K. We wll llusre hs by some exmples. Exmple..5: Le V = V V = b,b,c,d N(Q) c d n x N(Q);0 = 0 be srong neurosophc blner sublgebr of V over he neurosophc feld QI. Tke W = W W = QI = x QI; 0,,..., = 0 V V, W s pseudo srong neurosophc blner sublgebr of V over he subfeld Q of N(Q). DEFINITION..8: Le V = V V be srong neurosophc blner lgebr over he neurosophc feld K. Le W = W W V be bvecor spce over he rel feld F K. We cll W = W W V V s pseudo bvecor subspce of V over he rel subfeld F of K. 40

42 We wll llusre hs by some smple exmples. Exmple..6: Le V = V V = b,b,c,d N(Q) c d = x N(Q); 0,, = 0 be neurosophc blner lgebr over he neurosophc feld N(Q). Tke W = W W = b,b Q x Q; = 0,,,...,8 = 0 V V, W s pseudo bvecor spce over he feld Q. Exmple..7: Le V = V V = b 0,b,c Z9I 0 0 c {(N(Z 9 ) N(Z 9 ) N(Z 9 ) N(Z 9 ))} be srong neurosophc blner lgebr over he neurosophc feld N(Z 9 ). Tke W = W W = b 0,b,c Z 0 0 c 9 {(Z 9 Z 9 {0} {0})} V V ; W s pseudo bvecor subspce of V over he feld Z 9. DEFINITION..9: Le V = V V be srong neurosophc blner lgebr over he neurosophc feld K. Le W = W 4

43 W V V be blner lgebr over rel subfeld F of K. We defne W o be pseudo blner sublgebr of V over he feld F. We wll llusre hs by some smple exmples. Exmple..8: Le V = V V = b,b,c,d N(Q) c d = x N(Q); 0,,, = 0 be srong neurosophc blner lgebr over he neurosophc feld N(Q). Tke W = W W = b,b,c,d Q c d = x Q; 0,,, = 0 V V ; W s pseudo blner sublgebr of V over Q. Exmple..9: Le V = V V = b c d 0 d e f,b,c,d,e,f,g,h, N(Z 7 ) 0 0 g h {(N(Z 7 ) N(Z 7 ) N(Z 7 ) N(Z 7 ) N(Z 7 ))} be srong neurosophc blner lgebr over he neurosophc feld N(Z 7 ). Tke W = W W = 4

44 b 0 0,b,c,d Z 0 0 c d 7 {Z 7 {0} Z 7 {0} Z 7 } V V ; W s pseudo blner sublgebr of V over he feld Z 7 N(Z 7 ). DEFINITION..0: Le V = V V be srong neurosophc bvecor spce over he neurosophc feld K. Le W = W W be srong neurosophc bvecor spce over he sme neurosophc feld K. Le T : V W.e., T = T T : V V W W be bmp such h T : V W s srong neurosophc lner rnsformon from V o W ; =,. We defne T = T T o be srong neurosophc lner brnsformon from V o W. If W = V hen we cll T o be srong neurosophc lner boperor on V. SNHom K (V, W) denoes he se of ll srong neurosophc lner brnsformons from V o W. SNHom K (V, V) denoes he se of ll srong neurosophc lner boperor from V o V. Ineresed reder s requesed o gve exmples. Also he sudy of subsrucure preservng srong neurosophc lner brnsformons (boperors) s n neresng feld of reserch. Now we proceed ono defne blnerly ndependen bvecors nd oher reled properes. DEFINITION..: Le V = V V be srong neurosophc bvecor spce over he neurosophc feld K. A proper bsubse S = S S V V s sd o be bbss of V f S s blnerly ndependen bse nd ech S V generes V ; h s S s bss of V rue for =,. 43

45 DEFINITION..: Le V = V V be srong neurosophc bvecor spce over he neurosophc feld K. Le X = X X V V be bse of V, we sy X s lnerly bndependen bsubse of V over K f ech of he subses X conned n V s lnerly ndependen subse of V over he K; =,. The reder s expeced o prove he followng: THEOREM..: Le V = V V be srong neurosophc bvecor spce over he neurosophc feld K. Le B = B B be bbss of V over K hen B s lnerly bndependen subse of V over K. If X = X X be bsubse of V whch s blnerly ndependen bsubse of V hen X n generl need no be bbss of V over K. We wll expln hs by some exmples. Exmple..0: Le V = V V = {(QI QI QI)} x 0 n ; QI = 0 be srong neurosophc bvecor spce over he neurosophc feld QI. Le B =B B = {(I, 0, 0), (0, I, 0), (0, 0, I)} {I, Ix, Ix,, Ix n,, Ix } V V be bbss of V over he neurosophc feld QI. Tke X = X X = {I, 0, I), (0, 3I, I)} {I, Ix, Ix, Ix 3, Ix 7 } V V ; X s lnerly ndependen bsubse of V bu s no bbss of V over QI. Exmple..: Le V = V V = c b d,b,c,d Z3I Z3I; 6 be srong neurosophc bvecor spce over he neurosophc feld Z 3 I. Le B = B B = 44

46 I 0 0 I ,,, I 0 0 I I I I,,, ,, I I I V V, B s bbss of V over Z 3 I. Tke X = I I 0 0, 0 0 I I 3I 0 I 0 I 4I 0 I 0,, I I 0 I 0 4I = X X V V, X s only lnerly ndependen bse of V bu s no bbss of V over Z 3 I. DEFINITION..3: Le V = V V be srong neurosophc bvecor spce over he neurosophc feld K. Le X = X X V V, f X s no blnerly ndependen bsubse of V hen we sy X s blnerly dependen bsubse of V. Exmple..: Le V = V V = {QI QI QI QI} 5 x QI;0 5 = 0 be srong neurosophc bvecor spce over he neurosophc feld QI. Le X = X X = {(I, I, 0, 0), (0, I, I, 0), (0, 0, I, I), (I, I, I, I), (3I, I, I, 0)} {I, Ix, + 3Ix 3, 5Ix 3 + 3Ix, Ix 5 + 3Ix + 45

47 5Ix + 3Ix 4 } V V. I s esly verfed X s lnerly dependen bsubse of V over QI. Exmple..3: Le V = V V = c b d,b,c,d ZI = 0 x 0 ; ZI be srong neurosophc blner lgebr over he neurosophc feld Z I. B = B B = I 0 0 I ,,, I 0 0 I {I, Ix, Ix,, Ix n, } s bbss of B. I I I 0 I I I I 0 I,,,, 0 I 0 0 I I I I 0 0 {I + Ix + Ix 3 + Ix, Ix, I, Ix, I + Ix } s lnerly dependen bsubse of V over Z I. The number of belemens n he bbss B = B B s he bdmenson of V = V V, denoed by B = ( B, B ). If B = ( B, B ) = (n, m) nd f n < nd m < hen we sy V s fne bdmensonl srong neurosophc blner lgebr (bvecor spce) over he neurosophc feld K. Even f one of m or n s or boh m nd n s nfne hen we sy he bdmenson of V s nfne. Exmple..4: Le V = V V = c b d,b,c,d QI {QI QI QI} be srong neurosophc bvecor spce over he neurosophc feld QI. B = B B = 46

48 I 0 0 I ,,, I 0 0 I {(I 0 0), (0, I, 0) (0, 0, I)} V V ; B s bbss of V over QI nd he bdmenson of V s fne (4, 3). Exmple..5: Le V = V V = x ZI {(Z I Z I)} = 0 be srong neurosophc bvecor spce over Z I. B = B B = {I, Ix, Ix,, Ix n, } {(I, 0), (0, I)} V V s bbss of V over Z I. The bdmenson of V s (, ). Exmple..6: Le V = V V = x QI; =,,..., = 0 c b d,b,c,d RI be srong neurosophc blner lgebr over he neurosophc feld QI. B = B B = {I, Ix, Ix,, Ix n, } {n nfne bss for V } s bbss of V over QI. Thus he bdmenson of V s nfne nd B = (, ). Exmple..7: Le V = V V = c b d,b,c,d QI {RI RI} 47

49 be srong neurosophc bvecor spce over he neurosophc feld QI. Tke B = B B = I 0 0 I ,,, I 0 0 I {An nfne se}, B s bbss of V over QI. The bdmenson of V s (4, ); hus he bdmenson of V s nfne. I s neresng o noe h f V nd W re srong neurosophc bvecor spces over he neurosophc feld K. Suppose bdmenson of V s (n, n ) hen we sy he bdmenson of V nd W re he sme f nd only f W s jus of bdmenson (n, n ) or (n, n ). DEFINITION..4: Le V = V V nd W = W W be wo srong neurosophc bvecor spces over he neurosophc feld K. Le T = T T be blner rnsformon (lner brnsformon) from V o W defned by T : V W j, =,, j =,, such h T : V W nd T : V W or T : V W nd T : V W. The bkernel of T denoed by kert = kert kert where ker T ={ν V T(ν ) = 0; =, }. Thus bker T = {(ν, ν ) V V / T(ν, ν ) = T (ν ) T(ν ) = 0 0}. I s esly verfed ker T s proper neurosophc bsubgroup of V. Furher ker T s srong neurosophc bsubspce of V. Exmple..8: Le V = V V = QI, QI, 8 48

50 be srong neurosophc bvecor spce over QI. W = W W = 7 x QI;0 7 = QI, be srong neurosophc bvecor spce over QI. Defne bmp T = T T : V V W W by T : V W nd T : V W such h 3 3 T = nd T = 5 x 6 = where 0,, 3, 4 3, 5 4, 6 5, 7 6 nd 8 7. T = T T s lner bmp. bker T = Ineresed reder cn consruc more exmples n whch bker T s proper non zero neurosophc bsubspce of V. We wll prove resuls when we defne srong neurosophc n-vecor spces n >, for n = gves he srong neurosophc bvecor spce. Furher neurosophc bvecor spces (blner lgebrs) 49

51 nd he srong neurosophc bvecor spces (blner lgebrs) whch we hve defned n secons. nd. re ype I neurosophc bvecor spces nd srong neurosophc bvecor spces respecvely. In he followng secon we defne ype II neurosophc bvecor spces (blner lgebrs)..3 Neurosophc Bvecor Spces of Type II In hs secon we proceed ono defne neurosophc bvecor spces of ype II nd neurosophc lner blgebrs (or blner lgebrs) of ype II. We dscuss severl neresng properes bou hem. We lso gve he dfference beween ype I nd ype II neurosophc bvecor spces. DEFINITION.3.: Le V = V V where V s neurosophc vecor spce over he rel feld F nd V s neurosophc vecor spce over he rel feld F such h F F, F F, F F nd V V, V V nd V V. We cll V o be neurosophc bvecor spce over he bfeld F = F F of ype II. We wll llusre hs by some smple exmples. Exmple.3.: Le V = V V where b V =,b,c,d Z7I c d be neurosophc vecor spce over he feld Z 7 nd V = 3 4 N(Q), s neurosophc vecor spce over he feld Q. V = V V s neurosophc bvecor spce over he bfeld F = Z 7 Q of ype II. 50

52 Exmple.3.: Le V = V V where V = {QI [x]} neurosophc vecor spce over he feld Q nd V = b 0 0 d e,b,c,d,e,f Z 0 0 f be neurosophc vecor spce over he feld Z. V = V V s neurosophc bvecor spce over he bfeld F = Q Z of ype II. Exmple.3.3: Le V = V V where V = {Z 3 I Z 3 I Z 3 I Z 3 I} s neurosophc vecor spce over he feld Z 3 nd V = Z 3, 8 be neurosophc vecor spce over he feld Z 3. V = V V s neurosophc bvecor spce over he bfeld F = Z 3 Z 3 of ype II. DEFINITION.3.: Le V = V V be neurosophc bvecor spce over he bfeld F = F F of ype II. Le W = W W V V, f W s neurosophc bvecor spce over he bfeld F = F F of ype II, hen we cll W o be neurosophc bvecor subspce of V over he bfeld F = F F of ype II. We wll llusre hs by exmples. Exmple.3.4: Le V = V V = c b d,b,c,d Z7I 5

53 ZI, 8 be neurosophc bvecor spce of V over he bfeld F = Z 7 Z of ype II. Tke W = W W = b,b Z 7 I ZI,=,,4,5,6,3 V V ; W s neurosophc bvecor subspce of V over he bfeld Z 7 Z of ype II. Exmple.3.5: Le V = V V = c b d,b,c,d QI {Z 3 I Z 3 I Z 3 I Z 3 I Z 3 I} be neurosophc bvecor spce over he bfeld F = Q Z 3 of ype II. Tke W = W W = QI {( ) Z 3 I} V V ; W s neurosophc bvecor subspce of V over he bfeld F = Q Z 3. Now we defne subsrucures on hese neurosophc bvecor spces over he bfeld. I s pernen o menon here h he erm ype II wll be suppressed s one cn esly undersnd by he very defnon s dsnc from ype I. 5

54 DEFINITION.3.3: Le V = V V be neurosophc bvecor spce over he rel bfeld F = F F. Le W = W W V V nd K = K K F F = F. If W s neurosophc bvecor spce over he bfeld K = K K hen we cll W o be specl subneurosophc bvecor subspce of V over he bsubfeld K of F. We wll gve n exmple of hs defnon. Exmple.3.6: Le V = V V = {Q(, 3)I Q(, 3)I} b,b,c,d Q( 5, 7) c d be neurosophc bvecor spce over he bfeld F = Q(, 3) Q( 5, 7) = F. Tke W = {Q( )I Q( )I} Q( 5) = W W V V, W s specl subneurosophc bvecor subspce of V over he subfeld Q( ) Q( 5) = K K Q(, 3) Q( 5, 7) = F. Now we defne he neurosophc bvecor spce V o be bsmple f V hs no proper specl subneurosophc bvecor subspce over bsubfeld. We wll llusre hs by some exmples. Exmple.3.7: Le V = V V = b {RI}, b,c,d Z I c d 7 53

55 be neurosophc bvecor spce over he rel bfeld F = Q Z 7. We see he rel bfeld s bsmple;.e., hs no subbfelds or bsubfelds. So V s bsmple neurosophc bvecor spce over F. Exmple.3.8: Le V = V V = {Z I Z I Z I Z I} b,b,c,d Z 3 I c d be neurosophc bvecor spce over he rel bfeld F = Z Z 3. V s bsmple neurosophc bvecor spce over F. We see boh Z nd Z 3 re prme felds of chrcersc wo nd hree respecvely. In vew of hs we hve he followng heorem. THEOREM.3.: Le V = V V be neurosophc bvecor spce over bfeld F = F F. If boh F nd F re prme felds hen V s bsmple neurosophc bvecor spce over he rel bfeld F = F F. The proof of he bove heorem s lef s n exercse o he reder. A nurl queson rse; f one of he felds F nd F lone s prme feld cn we hve some specl ype of subsrucures. In vew of hs we hve he followng defnon. DEFINITION.3.4: Le V = V V be neurosophc bvecor spce over he rel bfeld F = F F where one of F or F s prme feld. Le W = W W be such h W s neurosophc vecor subspce of V over K F (F s prme feld ) nd W s neurosophc vecor subspce of V over F ; hen we cll W = W W o be qus specl neurosophc bvecor subspce of V over he qus bsubfeld K F. (If F = F F s bfeld, K F s proper subfeld of F hen K F s clled he qus bsubfeld of he bfeld F = F F ). We wll llusre hs by some exmples. 54

56 Exmple.3.9: Le V = V V = b,b,c,d Z 7 I c d {RI RI RI} be neurosophc bvecor spce over he bfeld F = Z 7 Q(, 3, 5, 7, ). Tke W = W W b,b,c,d Z I c d = 7 {RI {0} RI} V V, W s qus specl neurosophc bvecor subspce of V over he qus bsubfeld Z 7 Q( ) of he bfeld F. Exmple.3.0: Le V = V V = {Z 7 I [x]} {RI RI RI} be neurosophc bvecor spce over he rel bfeld Z 7 R. Le W = W W = 9 x 0 9; Z 7 I} {QI QI QI} = 0 V V ; W s qus specl neurosophc bvecor subspce of V over he rel qus bfeld Z 7 Q Z 7 R. Now we proceed on o defne he noon of bbss of he neurosophc bvecor spce of ype II. DEFINITION.3.5: Le V = V V be neurosophc bvecor spce of ype II over he bfeld F = F F. Le B = B B V V be bsubse of V such h B s lnerly ndependen bsubse of V ; nd generes V for =,, hen we cll B o be bbss of V over he bfeld F F = F. We wll llusre hs by some smple exmples. 55

57 Exmple.3.: Le V = V V = Z7I; Z I; 8 be neurosophc bvecor spce of ype II over he bfeld F = Z 7 Z 5. Tke B = B B = I I I,,, ,, I I I I 0 0 I I 0 0 I,,,, ,,, I 0 0 I I 0 0 I V V, B s bbss of V over he bfeld. 56

58 Exmple.3.: Le V = V V = QI Z I 37 be neurosophc bvecor spce over he bfeld F = F F = Q Z 37. Tke B = B B = I I I I I I I I I I I I V V, B s bbss of V over he bfeld F = Q Z 37. DEFINITION.3.6: Le V = V V be neurosophc bvecor spce over he bfeld F = F F. Le P = P P V V be proper bsubse of V such h ech P s lnerly ndependen subse of V over F ; =, ; hen we defne P = P P o be blnerly ndependen bsubse of V over he bfeld F = F F or P s defned o be he lnerly bndependen bsubse of V over he bfeld F = F F. I s neresng nd mporn o noe h every bbss s lnerly bndependen bsubse, bu lnerly bndependen bsubse need no n generl o be bbss of V over he bfeld F. We wll llusre hs suon by n exmple. Exmple.3.3: Le V = V V = 57

59 b,b,c,d RI c d 3,b ZI; 7 3 b b b3 be neurosophc bvecor spce of ype II over he bfeld F = Q Z 7. Tke B = B B = I 0 0 I ,,,, I 0 0 I I I I I I I,, I I I V V. Clerly B s lnerly bndependen bsubse of V bu s no bbss of V. Thus n generl every lnerly bndependen bsubse of V need no be bbss of V. DEFINITION.3.7: Le V = V V be neurosophc bvecor spce over he bfeld F = F F, nd W = W W be noher neurosophc bvecor spce over he sme bfeld F = F F h s V nd W re vecor spces over he feld F, =,. Le T = T T be bmp from V o W; where T : V W s lner rnsformon from V o W, =,. We defne T = T T : V = V V W = W W o be neurosophc lner brnsformon of V o W of ype II. We wll llusre hs by smple exmple. Exmple.3.4: Le V = V V = b,b,c,d Z 7 I c d 58

60 Z3I; 8 be neurosophc bvecor spce of ype II over he bfeld F = F F = Z 7 Z 3. Le W = W W = {Z 7 I Z 7 I Z 7 I Z 7 I} Z3I; be neurosophc bvecor spce of ype II over he bfeld F = F F = Z 7 Z 3. Defne T = T T : V = V V W = W W where T : V W nd T : V W s defned by b T = (, b, c, d) c d nd T = I s esly verfed T s lner brnsformon of V o W. Noe: If we ke n he defnon.3.7; W = V hen we cll T o be lner boperor on V of ype II. We wll denoe by NHom(V,W), he collecon of ll neurosophc lner F F brnsformons of V o W. NHom(V,V) F F denoes he collecon of ll neurosophc lner boperors of V o V. 59

61 Exmple.3.5: Le V = V V = b,b,c,d QI c d Z9I; 0 ; be neurosophc bvecor spce over he bfeld F = Q Z 9. Defne T = T T : V = V V V = V V where T : V V nd T : V V such h, nd b T = c d T where, b Z 9 I. = b b b b b I s esly verfed T s neurosophc blner operor on V of ype II. DEFINITION.3.8: Le V = V V nd W = W W be wo neurosopc bvecor spces over he bfeld F = F F. Le T = T T : V V = V W W = W be lner brnsformon of V o W. The bkernel of T denoed by ker T = ker T ker T where ker T = { v V T ( v ) = 0 } ; =,. Thus ker T = {(v, v ) V V / T(v, v )} = {T (v ) T (v ) = 0 0}. I s esly verfed h ker T s proper neurosophc bsubgroup of V. Fuher ker T s neurosophc bsubspce of V. The reder s expeced o gve some exmples. 60

62 THEOREM.3.: Le V = V V nd W = W W be wo neurosophc bvecor spces over he bfeld F = F F of ype II nd suppose V s fne bdmensonl. Le T = T T be neurosophc blner rnsformon (lner brnsformon) of V no W. (T : V W ; =, ). Then brnk T + bnully T = (n, n ) dm V = bdmenson V; h s (brnk T =) rnk T rnk T + (bnully T =) nully T nully T = (bdmenson V = ) dmv dm V = (n, n ). (Here dm V = n ; =, ). The proof s lef s n exercse o he reder. Furher he followng heorem s lso lef s n exercse o he reder. THEOREM.3.3: Le V = V V nd W = W W be wo neurosophc bvecor spces over he bfeld F = F F. Le T = T T nd S = S S be neurosophc blner rnsformons from V no W. The bfuncon (T + S) = (T T + S S ) = (T + S ) (T + S ) s defned by (T + S)(α) = (T + S ) (T + S ) (α α ) = (T + S ) α (T + S )α = (T α + S α ) (T α + S α ) s neurosophc lner brnsformon from V = V V o W W. (α = α α V V ). If C = C C s bsclr from he bfeld hen he bfuncon (CT)α = (C C ) (T T ) (α α ) = C T α C T α s blner rnsformon (lner brnsformon ) from V no W. Thus he se of ll lner brnsfomons defned by bddon nd sclr bmulplcon s neurosophc 6

63 bvecor spce (vecor bspce) over he sme bfeld F = F F. Le NL(V, W) = NL (V, W ) NL (V, W ) be neurosophc bvecor spce over he bfeld F = F F. Furher f V = V V s neurosophc bvecor spce over he bfeld F = F F of fne bdmenson (n, n ) nd W = W W s neurosophc bvecor spce of fne dmenson (m, m ) over he sme bfeld F = F F. Then NL(V, W) s of fne bdmenson nd hs (m n, m n ) bdmenson over he sme bfeld F = F F. Furher we hve noher neresng propery bou hese neurosophc bvecor spces. Le V = V V, W = W W nd Z = Z Z be hree neurosophc bvecor spces over he sme bfeld F = F F. Le T be neurosophc blner rnsformon from V no W nd S be neurosophc lner brnsformon from W no Z. Then he bcomposed bfuncon S o T = ST defned by ST(α) = S(T(α)) s neurosophc blner rnsformon from V no Z. The reder s expeced o prove he bove clm. Now we proceed on o defne he noon of neurosophc blner lgebr or neurosophc lner blgebr of ype II over he bfeld F = F F. DEFINITION.3.9: Le V = V V be neurosophc bvecor spce of ype II over he bfeld F = F F. If ech V s neurosophc lner lgebr over F, =,, hen we cll V o be neurosophc blner lgebr over he bfeld F = F F of ype II. We wll llusre hs by some smple exmples. Exmple.3.6: Le V = b,b,c,d QI c d 6

64 {(, b, c, d, e), b, c, d, e Z 7 I} be neurosophc bvecor spce over he bfeld F = Q Z 7. V s clerly neurosophc blner lgebr over F. Exmple.3.7: Le V = V V = ZI; {Z 3 I[x]; ll polynomls n he vrble x wh coeffcens from Z 3 I}; V s neurosophc blner lgebr over he bfeld F = Z Z 3. Exmple.3.8: Le V = V V = 3 4 Z7I; b,b,c,d QI ; c d V s only neurosophc bvecor spce over he bfeld Z 7 Q. Clerly V s no neurosophc blner lgebr over he bfeld of ype II s V s no neurosophc lner lgebr over he feld Z 7. Thus we hve he followng neresng resul, he proof of whch s lef s n exercse for he reder. THEOREM.3.4: Le V = V V be neurosophc blner lgebr over bfeld F = F F of ype II. Clerly V s neurosophc bvecor spce over he bfeld F. However neurosophc bvecor spce of ype II need no n generl be neurosophc blner lgebr of ype II. 63

65 Now we proceed on o defne he new noon of neurosophc lner bsublgebr or neurosophc blner sub lgebr of ype II DEFINITION.3.0: Le V = V V be neurosophc blner lgebr over bfeld F = F F of ype II. Tke W = W W V V ; W s neurosophc sub blner lgebr or neurosophc blner sublgebr of V f W s self neurosophc lner blgebr of ype II over he bfeld F = F F. We wll llusre hs suon by some exmples. Exmple.3.9: Le V = V V = b,b,c,d QI c d {( ) Z I; 6} be neurosophc blner lgebr of ype II over he bfeld F = Q Z. Tke W = W W = QI {( ) Z I} V V, W s neurosophc blner sublgebr of V over he bfeld F = Q Z of ype II. Exmple.3.0: Le V = V V = Z3I;

66 {QI[x]; ll polynomls n he vrble x wh coeffcens from QI} be neurosophc blner lgebr of ype II over he bfeld F = Z 3 Q. Tke W = W W = Z3I; {All polynomls of even degree wh coeffcens from he feld QI} V V ; W s neurosophc blner sublgebr of V of ype II over he bfeld Z 3 Q. DEFINITION.3.: Le V = V V be neurosophc blner lgebr over bfeld F = F F of ype II. Le W = W W V V, suppose W s only neurosophc bvecor spce of ype II over he bfeld F = F F nd s no neurosophc blner sublgebr of V of ype II over he bfeld F hen we sy W s pseudo neurosophc bvecor subspce of V over he bfeld F = F F of ype II. We wll llusre hs by some exmples. Exmple.3.: Le V = b,b,c,d QI c d {Z 7 I[x]; ll polynomls n he vrble x wh coeffcens from Z 7 I} be neurosophc blner lgebr over he bfeld F = F F = Q Z 7. Tke W = W W = 0 b 0,b QI 0 = 0 x ZI;