Chapter 2 Introduction to Algebra. Dr. Chih-Peng Li ( 李 )

Transcription

1 Chpter Introducton to Algebr Dr. Chh-Peng L 李

2 Outlne Groups Felds Bnry Feld Arthetc Constructon of Glos Feld Bsc Propertes of Glos Feld Coputtons Usng Glos Feld Arthetc Vector Spces

3 Groups 3

4 Let G be set of eleents. Groups A bnry operton * on G s rule tht ssgn to ech pr of eleents nd b unquely defned thrd eleent c*b n G. Defnton: A group s set G wth bnry opertor * tht stsfes the followng condtons : Closure: b, G b G Assoctve: bc,, G * b* c * b * c Identty: e G s.t. G, * e e* Ths eleent e s clled n dentty eleent of G. Inverse: G, ' G, s.t. * ' '* e The eleent s clled n nverse of. 4

5 Groups Theore.: The dentty eleent n group G s unque. pf : e e*e e *e e fro defnton Theore.: The nverse of group eleent s unque. pf : *e dentty ** nverse ** ssoctve e* nverse dentty 5

6 Rerks: 6 Groups A group G s sd to be couttve beln group f, b G, *b b* Order: The nuber of eleents n group. We denote t G. Fnte group: A group of fnte order. Eples of couttve groups: ntegers under rtonl nubers under {,} under * rel-vlued tr under {,,,., -} under odulo- ddton s couttve see eple. {,, 3.p-} under odulo- p ultplctor p s pre s lso couttve see eple.3

7 Groups Eple. Consder the set of two ntegers, G {, }. Let us defne bnry operton, denoted by, on G s follows : Ths bnry operton s clled odulo- ddton. The set G {, } s group under odulo- ddton. It follows fro the defnton of odulo- ddton tht G s close under nd s the dentty eleent. The nverse of s tself nd the nverse of s lso tself. Thus, G together wth s couttve group. 7

8 Groups Eple. Let be postve nteger. Consder the set of nteger G {,,,, -}. Let denote rel ddton. Defne bnry operton on G s follows: For ny ntegers nd j n G, j r, where r s the render resultng fro dvdng j by. The render r s n nteger between nd - Eucld s dvson lgorth nd s therefore n G. Hence G s closed under the bnry operton, clled odulo- ddton. Frst we see tht s the dentty eleent. For < <, nd re both n G. Snce 8

9 Groups It follows fro the defnton of odulo- ddton tht Therefore, nd - re nverses to ech other wth respect to. It s lso cler tht the nverse of s tself. Snce rel ddton s couttve, t follows fro the defnton of odulo- ddton tht, for ny ntegers nd j n G, j j. Therefore odulo- ddton s couttve. Net we show tht odulo- ddton s lso ssoctve. Let, j, nd k be three ntegers n G. Snce rel ddton s ssoctve, we hve j k j k j k 9

10 Groups Dvdng j k by, we obtn j k q r, where q nd r re the quotent nd the render, respectvely, nd r <. Now, dvdng j by, we hve j q r, wth r <. Therefore, j r. Dvdng r k by, we obtn r k q r wth r <. Hence r k r nd j k r. Cobnng. nd., we hve j k q q r, Ths ples tht r s lso the render when j k s dvded by. Snce the render resultng fro dvdng n nteger by nother nteger s unque, we ust hve r r.

11 Groups As result, we hve j k r. Slrly, we cn show tht j k r. Therefore j k j k nd odulo- ddton s ssoctve. Ths concludes our proof tht the set G {,,,, -} s group under odulo- ddton. We shll cll ths group n ddtve group.

12 Groups Eple.3: Let p be pre e.g. p, 3, 5, 7,,. Consder the set of ntegers, G {,,,, p-}. Let denote rel ultplcton. Defne bnry operton $ on G s follows: For nd j n G, $ j r, where r s render resultng fro dvdng j by p. The set G {,,,, p-} s group under odulo-p ultplcton. Frst we note tht j s not dvsble by p. Hence < r < p nd r s n eleent n G. Therefore, the set G s closed under the bnry operton $, referred to s odulo-p ultplcton. We cn esly check tht odulo-p ultplcton s couttve nd ssoctve. The dentty eleent s. The only thng left to be proved s tht every eleent n G hs n nverse.

13 Groups Let be n eleent n G. Snce p s pre nd <p, nd p ust be reltvely pre.e. nd p don t hve ny coon fctor gret thn. It s well known tht there est two ntegers nd b such tht b p.3 nd nd p re reltvely pre Eucld s theore. Rerrngng.3, we hve - b p..4 Ths sys tht when s dvded by p, the render s. If < < p, s n G nd t follows fro.4 nd the defnton of odulo-p ultplcton tht $ $. 3

14 4 Groups Therefore s the nverse of. However, f s not n G, we dvde by p, q p r. Snce nd p re reltvely pre, the render r cnnot be nd t ust be between nd p-. Therefore r s n G. Now cobnng.4 nd.5, we obtn r - b qp. Therefore r $ $ r nd r s the nverse of. Hence ny eleent n G hs n nverse wth respect to odulo-p ultplcton. The group G {,,,, p-} under odulo-p ultplcton s clled ultplctve group. Def: Let H G & H φ epty set, then H s sd to be subgroup of G f H s group.

15 Felds 5

16 Felds Roughly spekng, feld s set of eleents n whch we cn do ddton, subtrcton, ultplcton, nd dvson wthout levng the set. Addton nd ultplcton ust stsfy the couttve, nd dstrbutve lws. Defnton: Let F be set of eleents on whch two bnry opertons, clled ddton nd ultplcton, re defned. The set F together wth the two bnry opertons nd s feld f the followng condtons re stsfed: F s couttve group under ddton. The dentty eleent wth respect to ddton s clled the zero eleent or the ddtve dentty of F nd s denoted by. 6

17 Felds The set of nonzero eleents n F s couttve group under ultplcton. The dentty eleent wth respect to ultplcton s clled the unt eleent or the ultplctve dentty of F nd s denoted by. Multplcton s dstrbutve over ddton; tht s, for ny three eleents, b, nd c n F, bc b c 7

18 8 Felds A feld conssts of t lest two eleents, the ddtve dentty nd the ultplctve dentty. The nuber of eleents n feld s clled the order of the feld. A feld wth fnte nuber of eleents s clled fnte feld. In feld, the ddtve nverse of n eleent s denoted by nd the ultplctve nverse of s denoted by - provded tht. Subtrctng feld eleent b fro nother feld eleent s defned s ddng the ddtve nverse b of b to. [-b -b]. If b s nonzero eleent, dvdng by b s defned s ultplyng by the ultplctve nverse b - of b. [/b b - ].

19 Felds Property I. For every eleent n feld,. Proof. Addng to both sdes of the equlty bove, we hve: - - Slrly, we cn show tht. Therefore, we obtn. Property II. For ny two nonzero eleents nd b n feld, b. Proof. Fro defnton, nonzero eleents of feld re closed under ultplcton. 9

20 Felds Property III. b nd ply tht b. Ths s drect consequence of Property II. Property IV. For ny two eleents nd b n feld, - b- b -b. b- b b- b - b ust be the ddtve nverse of b nd b- b. Slrly, we cn prove tht b -b. Property V. For, b c ples tht bc. Snce s nonzero eleent n the feld, t hs ultplctve nverse -. Multplyng both sde of b c by -, we obtn - b - c - b - c b c > bc

21 Felds Soe eples: R rel nuber set C cople nuber Q Rtonl nuber GFq ests f q, p s pre p nfnte felds E.4. Bnry feld GF wth odulo- ddton odulo- ultplcton

22 Felds E.5: GFp, p s pre. Pre Feld {,,,.p-} s n beln group under odulo-p ddton. {,, p-} s n ben group under odulo-p ultplcton. Fct: rel nuber ultplcton s dstrbutve over rel nuber ddton. Ths ples tht odulo-p ultplcton s dstrbutve over odulo-p ddton. {,,,.p-} s feld of order p under odulo-p ddton nd ultplcton. In fct, for ny postve nteger, t s possble to etend the pre feld GFp to feld of p eleents clled n etenson feld of GFp nd s denoted by GFp. Furtherore, the order of ny fnte feld s power of pre. Fnte felds re lso clled Glos feld.

23 Felds Def: Chrcterstc of GFq Consder the unt eleent n GFq; sllest postve λ nteger λ s.t., then λ s clled the chrcterstc of GFq EX. The chrcterstc of GF s The chrcterstc of GFp s p k k for k < p, p 3

24 Felds Theore.3: The chrcterstc of GFq s pre. pf λ k k, < λ k k λ. k or contrdcts the defnton of Fct: For ny two dstnct postve nteger k, < k λ 4

25 Rerks:,,,, λ λ q λ λ Felds re dstnct eleents n GFq, whch for subfeld GF of GFq If, then q s power of proven n lter λ Def: Order of feld eleent n Let GFq,, sllest postve nteger n s.t.. n s clled the order of the feld eleent. 5

26 Rerks: Felds 3 n n,,,,, re ll dstnct, whch for group under the ultplcton of GFq. pf: Closure f j n, j j f j > n, we hve j n r where r n j j n r r n Inverse For < n, s the ultplctve nverse of Snce the powers of re nonzero eleents n GFq, they stsfy the ssoctve nd counttve lws. 6

27 7 Felds Def: Cyclc A Group s sd to be cyclc, f there ests n eleent n the group whose powers consttute the whole group. Theore.4: GF q nd, then q pf: Let b, b, b q- be the q- nonzero eleent of GFq, then b, b, b q- re lso nonzero nd dstnct. Thus, b b.. bq- b b bq- q- b b b q- b b b q- q- Theore.5: GF q nd, n s the order of then pf : If not, q- kn r <r<n q kn r n k r r n q

28 Def: Prtve GF q& Rerks:, s sd to be prtve f the order of s q- The powers of prtve eleent generte ll the nonzero eleents of GFq Every fnte feld hs prtve eleent. prob..7 Eple. GF5 ' 3 4 5, 4, 3,, s prtve eleents. prtve eleents re useful for constructng felds. Eple. GF7 3 s prtve eleent, the order of 4 s 3, whch dvdes 6. 8 Felds

29 Bnry Feld Arthetc 9

30 3 Bnry Feld Arthetc Sets of equtons e.g. XY, XZ, XYZ Solved by Grer s rule y z

31 Bnry Feld Arthetc Polynols over GF. We denote t GF[X]. Def: n f f f... f n f GF f f n, deg[f] n f f... f, f n, deg[f] Rerks: Polynols over GF wth degree e :, Polynols over GF wth degree e :,,, n In generl, wth degree n we hve polynols. 3

32 3 Added or subtrcted Multpled If g, then f Couttve n n f f g f g f g f g f... n g g g g n n o f g c c c c f g f g f g, n n g f c g f c f g g f f g g f Bnry Feld Arthetc

33 Assoctve f [g h] [f g] h f [g h] [f g] h Dstrbutve f [g h] [f g] [f h] Bnry Feld Arthetc Eucld s dvson lgorth Suppose deg[g], q, r GF[] s.t. fqg r, where deg[r] < deg[g] q : quotent, r : render e.g. If r, f s dvsble by g. [g dvdes f, g f] 33

34 Bnry Feld Arthetc Root GF, f f, then f s dvsble by e.g. f f, f s dvsble by e. Def: Irreducble p GF [] wth deg[p] s sd to be rreducble over GF f p s not dvsble by ny polynol over GF of degree less thn but greter thn zero. 34

35 e.g. ong, Bnry Feld Arthetc,,, only s n rreducble polynol wth degree. 3 s lso n rreducble poly wth degree 3. In generl, for ny, there ests n rreducble polynol of degree. Theore.6: Any rreducble polynol over GF of degree dvdes. See Theore.4 e.g

36 Bnry Feld Arthetc Def: Prtve An rreducble polynol p of degree s sd to be prtve n f the sllest postve nteger n for whch p dvdes s n. 4 5 e.g. p n but p! for n < 5 prtve p 5 t cn lso p not prtve Rerks: For gven >, there y be ore thn one prtve polynols of degree n. Lsts of prtve polynols see p.9 Tble.7 36

37 Bnry Feld Arthetc FACT: pf: f GF[ ], [ f ] f N n f f f... f n n f f... fn f f n f f... f n f f n... f n f... f... n f f f... f f n f n n 37

38 Constructon of Glos Feld 38

39 Constructon of Glos Feld Consder, n GF nd new sybol. Defne s follows:,, 3 j j tes 39

40 Constructon of Glos Feld j j j j j j j j { j,,,...,...} F wth be soete denoted by 4

41 Let prtve polynol Wth deg [p] & Snce p Constructon of Glos Feld Therefore, under the condton tht p GF[ ] ssue p q p q p q p F s fnte.e. F F { },,,,..., * * Fro, F s closed under 4

42 Constructon of Glos Feld * FACT: The nonzero eleents of F for couttve group wth order under * * Now defne n ddtve operton on F s.t. F fors couttve group under For <, g & over GF s.t. g p, where....e. deg [ ] 4

43 43 Constructon of Glos Feld FACT: For nd p re reltve pre s not dvsble by p For pf: If <j however, p s prtve polynol of degrees whch,, nd, j j < j j [ ] p g g j j j [ ] p g g j j j p p

44 Constructon of Glos Feld Therefore,,,,... re dstnct nonzero poly. of degree - or less. Recll tht g p n GF Replcng by n p / for n < q.e. { } nonzero eleents re represented by dstnct nonzero poly. of over GF wth degree - or less.... o, 44

45 45 Constructon of Glos Feld Snce zero eleent n y be represented by the zero poly. eleents n re represented by dstnct poly. of over GF wth degree - or less nd re regrded s dstnct eleents. Defne s follows:.. for So, * F * F j <,,..., j... j j... j jo o,, j

46 Constructon of Glos Feld * FACT: F s couttve group under ddtve dentty ddtve nverse couttve ssoctve { } FACT: * F,,,..., s Glos feld of eleents. * pf: s couttve group under F * s couttve group under F {} 46

47 Rerks: Constructon of Glos Feld * nd defned on F GF ply odulo- ddton nd ultplcton. {,} fors subfeld of GF.e. GF s clled the ground feld of GF The chrcterstc of GF s Two representtons for the nonzero eleent of GF the power representton for ultplcton the polynol representton for ddton s prtve eleent of GF 47

48 48 Constructon of Glos Feld Eple: over poly. prtve s 4 4 GF p p Set

49 49 Constructon of Glos Feld, Another useful representton of feld eleents n GF,..., tuple... : β

50 5 Constructon of Glos Feld E. Construct GF4 fro GF wth { } { },,,,,, p

51 Bsc Propertes of Glos Feld 5

52 5 Bsc Propertes of Glos Feld over root of the s GF p over of re roots,,, ] ][ [ GF p

53 Bsc Propertes of Glos Feld 4 3 p s rredeucble over GF.e. t doesn't hve ny 4 root over GF. However,t hs four roots over GF Theore.7: f GF [ ], β n etenson feld of GF If f.e. f β, then f β β s root, then β re ll roots pf: [ f ] f f β [ f β ] 53

54 Rerks: The Bsc Propertes of Glos Feld Let f GF[], f β s clled conjugte of β β GF β GF nd f β e.g f Then f... where s.t. f β, then GF GF 4 4,gven by Tble.8 4 The conjugtes of re roots of f FACT: If pf : β β GF 8 :,, re ll roots of nd β fro Theore.4, then β s 5 f, besdes root of & 54

55 Rerks: Bsc Propertes of Glos Feld All nonzero eleents of GF fro ll the roots of All eleents of GF for ll the roots of Def: nl polynol Let β GF, the poly. φ of sllest degree over φ β s clled the nl poly.of β. Rerks: β s root of β y be root of p GF[ ] wth degree[ p ] < GF s.t. 55

56 Bsc Propertes of Glos Feld The nl poly.of s The nl poly.of s The nl poly.of 7 s 4 3 Q : The nl poly.of 4 s or not? Theore.9: The nl poly. φ of feld eleent s rreducble. β pf: If not φ φ φ when < deg[ φ ] < deg[ φ] for, 56

57 Bsc Propertes of Glos Feld φ β φ β φ β φ β or φ β φ s not nl poly. FACT: f GF[ ]. Let φ be the nl poly. of β. If f β, then φ f pf: f φ r, deg[ r ] < deg[ φ ] f β φ β r β If r, then φ s not nl poly.of β. 57

58 Bsc Propertes of Glos Feld FACT: The nl poly. φ of β GF dvdes Rerk: ll the roots of φ re fro GF ccordng to Corollry.8. FACT: f GF[ ] nd f s n rreducble poly. Let β GF nd φ be the nl poly. of β. If f β, then φ f pf: fro fct φ & f φ f s rreducble φ f 58

59 Rerks: Bsc Propertes of Glos Feld Ths fct ples tht n rreducble poly. f wth root β s the nl poly. φ of β. Fro prevous Th.7 β, β, β,, β, re roots of φ. Let e be the sllest e e nteger s.t. β β, then β, β, β β re ll the dstnct conjugtes of β see prob..4 59

60 6 Bsc Propertes of Glos Feld FACT: s.t., then s n rreducble poly over GF pf: proof tht by frst proof be the sllest & e GF β β β e f e β Π [ ] GF f ] [ f f ] [ β β f e e Π Π Π β β β e Π β e

61 6 Bsc Propertes of Glos Feld Let β e Π β e Π f e β β e Π e f e f f f... Epnd where f e [ ] e e f f f f f e e e j j e j j f f f...

62 Bsc Propertes of Glos Feld f e f... fro & Ths holds only when or f f f f GF [ ] e prove tht f s rreducble over GF f not, f b & f β β b β β or b β, f β, hs e roots ββ,, β. Theore.7 deg [ ] e nd f f b β, the sereson s.t. b f Therefore, f ust be rreducble. 6

63 Th.4: Bsc Propertes of Glos Feld Let φ be the nl poly. of β GF nd e be the φ sllest e nteger β followng fro Th. &.3 e s.t. β β, Then e.g. β 3 GF 4 gven by Tble β, β, β 3 The nl poly. of β s φ

64 64 Bsc Propertes of Glos Feld e.g. Fnd the nl poly. Usng poly. representton for of φ n 4 7 GF r ,, r r r Hence φ r r r r r φ 4 3,r,r r, r

65 Bsc Propertes of Glos Feld All the nl poly. of eleents n GF 4. See p.38 Tble.9 Th.5: Let φ,, φ be the nl poly. of β GF & deg[ φ ] e. Then e be the sllest nteger s.t. β β. Moreover e drect fro Th.4&.5 Rerks: f be the nl poly. of then e proof s otted β GF wth deg[ f] e 65

66 Bsc Propertes of Glos Feld Mnl poly. of β GF 66 to see Append B Th.6: β GF, f β s prtve eleent of GF, then ll ts conjugtes eleent of GF re lso prtve pf: Let n be the order of β for >, then n n β β. fro Th.5. n β s prtve eleent of ts order s For β n -&, - n β, β GF re reltve pre,,

67 Bsc Propertes of Glos Feld - n fro, n β s lso prtve eleent. Eple: β 7 GF 4 gven by Tble β, β, β,β β β , β β 7 Clerly, s prtve eleent of GF β, β, β re ll prtve eleents of GF 4 67

68 Bsc Propertes of Glos Feld Th.7: nd hs order n, then ll the conjugte hve the se order n. See prob..5 β GF β 68

69 Coputtons Usng Glos Feld Arthetc 69

70 Coputtons Usng Glos Feld Arthetc 7 EX: Consder Y over GF 4 8 Y 4 X Y

71 Coputtons Usng Glos Feld Arthetc 7 EX: Solve f Tble.8 try nd error 6 f 3 over GF 4 gven by f 7 6 f 6, 7

72 Vector Spces 7

73 Vector Spces Def: V be set of eleents wth bnry operton s defned. F be feld. A ultplcton opertor between F nd v V s lso defned. The V s clled vector spce over the feld F f: V s couttve group under. F & v V v V dstrbutve lw u,v V nd,b F Assoctve Lw. v v u v b v u v b v b b v v v 73

74 Vector Spces The eleents of V re clled vectors nd the eleents of the feld F re clled sclrs. The ddton on V s clled vector ddton nd the ultplcton tht cobnes sclr n F nd vector n V s referred to s sclr ultplcton or product The ddtve dentty of V s denoted by. Property I. Let be the zero eleent of the feld F. For ny vector v n V, v. Property II. For ny sclr c n F, c. Left s n eercse 74

75 Vector Spces Property III. For ny sclr c n F nd ny vector v n V, -c v c -v -c v.e., -c v or c -v s the ddtve nverse of the vector c v. Left s n eercse Consder n ordered sequence of n coponents,,,..., n, where ech coponent s n eleent fro the bnry feld GF.e., or. Ths sequence s clled n n-tuple over GF. n Snce there re two choces for ech, we cn construct dstnct n-tuples. Let V n denote ths set. Now we defne n ddton on V n s followng : For ny u u,,..., nd u u n v v, v,..., v n n V n, u v u v, u v,..., u n vn.7 75

76 Vector Spces where uv s crred out n odulo- ddton. Clerly, u v s lso n n-tuple over GF. Hence V n s closed under the ddton. We cn redly verfy tht V n s couttve group under the ddton defned by.7. we see tht ll zero n-tuple,,, s the ddtve dentty. For ny v n V n, v v v,,, v, v v,..., v n vn Hence, the ddtve nverse of ech n-tuples n V n s tself. Snce odulo- ddton s couttve nd ssoctve, the ddton s lso couttve nd ssoctve. Therefore, s couttve group under the ddton. V n we defned sclr ultplcton of n n-tuple v n V n 76

77 Vector Spces by n eleent fro GF s follows : v, v,..., v n v, v,..., vn.8 where v s crred out n odulo- ultplcton. Clerly, v, v,..., v n s lso n n-tuple n Vn. If, v, v,..., v n v, v,..., vn v, v,..., V n v n By.7 nd.8, the set of ll n-tuples over GF fors vector spce over GF 77

78 Vector Spces Eple Let n. The vector spce V of ll -tuples over GF conssts of the followng 4 vectors : The vector su of nd s Usng the rule of sclr ultplcton defned by.8, we get V beng vector spce of ll n-tuples over ny feld F, t y hppen tht subset S of V s lso vector spce over F. Such subset s clled subspce of V. 78

79 Vector Spces Theore.8 Let S be nonepty subset of vector spce V over feld F. Then S s subspce of V f the followng condtons re stsfed : For ny two vectors u nd v n S, u v s lso vector n S. For n eleent n F nd ny vector u n S, u s lso n S. pf. Condtons nd sy sply tht S s closed under vector ddton nd sclr ultplcton of V. Condton ensures tht, for ny vector v n S, ts ddtve nverse - v s lso n S. Then, v - v s lso n S. Therefore, S s subgroup of V. Snce the vectors of S re lso vectors of V, the ssoctve nd dstrbutve lws ust hold for S. Hence, S s vector spce over F nd s subspce of V. 79

80 Vector Spces Let v, v,,vk be k vectors n vector spce V over feld F. Let,,, k be k sclrs fro F. The su v v kvk s clled lner cobnton of v, v,,vk. Clerly, the su of two lner cobntons of v, v,,vk, v v kvk bv bv bkvk b v b v kbk vk s lso lner cobnton of v, v,,vk, nd the product of sclr c n F nd lner cobnton of v, v,,vk, c v v kvk c v c v... c k v k s lso lner cobnton of v, v,,vk Theore.9 Let v, v,,vk be k vectors n vector spce V over feld F. The set of ll lner cobntons of v, v,,vk fors subspce of V. 8

81 Vector Spces A set of vectors v, v,,vk n vector spce V over feld F s sd to be lnerly dependent f nd only f there et k sclrs,,, k fro F, not ll zeros, such tht v v kvk A set of vectors v, v,,vk s sd to be lnerly ndependent f t s not lnerly dependent. Tht s, f v, v,,vk re lnerly ndependent, then v v kvk unless k. EX. The vectors,, nd re lnerly dependent snce 8

82 Vector Spces However,,, nd re lnerly ndependent. A set of vectors s sd to spn vector spce V f every vector n V s lner cobnton of the vectors n the set. In ny vector spce or subspce there ets t lest one set B of lnerly ndependent vectors whch spn the spce. Ths set s clled bss or bse of the vector spce. The nuber of vectors n bss of vector spce s clled the denson of the vector spce. Note tht the nuber of vectors n ny two bses re the se. 8

83 Vector Spces Consder the vector spce of ll n-tuples over GF. Let us for the followng nn-tuples : e e V n e n-..., where the n-tuple e hs only nonzero coponent t th poston. Then every n-tuple,,..., n n V n cn be epressed s lner cobnton of e, e,,en- s follows :,,..., n e e... n en 83

84 Vector Spces Therefore, e, e,,en- spn the vector spce of ll n-tuples over GF. We lso see tht e, e,,en- re lnerly ndependent. Let u u, u,..., u n nd v v, v,..., v n be two n-tuples n. We defne the nner product or dot product of u nd v s V n where u v nd u v u v re crred out n odulo- ultplcton nd ddton. Hence the nner product u v s sclr n GF. If u v, u nd v re sd to be orthogonl to ech other. The nner product hs the followng propertes : u v v u u vw u v u w u v u v u v uv uv... u n vn, 84 V n

85 Vector Spces Let S be k-denson subspce of nd let Sd be the set of vectors n V n such tht, for ny u n S nd v n Sd, u v. The set Sd contns t lest the ll-zero n-tuple,,,, snce for ny u n S, u. Thus, Sd s nonepty. For ny eleent n GF nd ny v n Sd, f v { v f Therefore, v s lso n Sd. Let v nd w be ny two vectors n Sd. For ny vector u n S, u vw u v u w. Ths sys tht f v nd w re orthogonl to u, the vector su v w s lso orthogonl to u. Consequently, v w s vector n Sd. It follows fro Theore.8 tht Sd s lso subspce of V n. Ths subspce s clled the null or dul spce of S. Conversely, S s lso the null spce of Sd. 85 V n

86 Vector Spces Theore. Let S be k-denson subspce of the vector spce Vn of ll n-tuples over GF. The denson of ts null spce Sd s n-k. In other words, ds dsd n. 86