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Bài tập đại số tuyến tính

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  1. PHAN HUY PHIJ • NGUYEN DOAN TUAN BAI TAP DAI SO TUYEN TINH NHA XUAT BAN HAI HOC QUOC GIA HA NOI
  2. Chin trach nhiem xual bcin Gicim doe: NGUYEN VAN THOA Tong bien Op: NGUYEN THIEF N GIAP Bien tap: HUY CHU DOAN 'MAN NGOC QUYEN Trinh bay Ilia: NGOC ANH BAI TAP DAI sq TUYEN TINH Ma s6: 01.249.0K.2002 In I .501) cudn, tai Xtiiing in NXI3 Giao thong van tai S6 xuat ban: 49/ 171/CXS. S6 Inch ngang 39 KH/XB In xong va Opt [Yu chi& CM/ I narn 2002.
  3. Lai NOI DAU Mon Dai s$ tuygn tinh dude dua vao giang day a hau hat cac trUnng dai hoc va cao dang nhtt 1a mot mon hoc cd se; can thigt d@ tigp thu nhUng mon hoc khan. Nham cung cap them mot tai lieu tham khao phut vu cho sinh vien nganh Toan vi cac nganh Ki thuat, chting Col Bien soan cugn "BM tap Dai so tuygn tinh". Cugn each dude chia lam ba chudng bao g6m nhUng van d6 Cd ban cna Dal so tuygn tinh: Dinh thfic va ma trail - Khong gian tuygn tinh, anh xa tuygn tinh, he phticing trinh tuygn tinh - Dang than phttdng. Trong mOi chudng chung toi trinh bay phan torn tat lY thuyat, cac vi du, cac hal tap W giai va cugi mOi chudng c6 phan hudng dan (HD) hoac dap s6 (DS). Cac vi du va bai tap &roc chon be a mac an to trung binh den kh6, c6 nhUng bai tap mang tinh 1± thuygt va nhUng bai tap ran luyen ki nang nham gain sinh vien higu sau them mon lice. Chung toi xin cam on Ban bien tap nha xugt ban Dai hoc Qugc gia Ha Nei da Lao digt, kien de cugn sach som dude ra mat ban doe. Mac du chting tea da sa dung 'Lai lieu nay nhigu narn cho sinh vien Toan Dal hoc Su pham Ha NOi va da co nhieu co gang khi bier, soon, nhUng chat than con cú khigm khuygt. Cluing toi rat mong nhan dude nhUng y kin clang gap cna dee gia. Ha N0i, thcing 3 !Lam 2001 NhOni bien soan 3
  4. rvikic LUC Chubhg .1: DINH THOC - MA TRA:N 7 A - Tom tat ly thuyeet 7 Đ1. Phep th6 7 Đ 2. Dinh thitc Đ 3. Ma tram 10 B - Vi dn 12 C - Bei tap 35 D HtiOng dein hoac clap so 43 Chudng 2. KHONG GIAN VECTO - ANH XA TUYEN TINH • PHUGNG TRINH TUYEN TINH 57 A - TOrn tat ly thuyeet 57 Đ1. Kh8ng gian vec to 57 Đ2. Anh xa tuyeen tinh 61 Đ 3. He phydng trinh tuy6n tinh 64 Đ4. Can true caa tai ding cku 67 B Vi dtt 71 C - Biti tap 96 Đ1. 'thong gian vec to va anh xa tuyeen tinh 96 Đ2. He pinking trinh tuy6n tinh 104 Đ3. Cau tit cna melt tu thing calu 106 D. Illidng sign ho(tc clap s6 110 5
  5. Đ1. Khong gian vec td va anh xn tuyin tinh 11( Đ 2. He phudng trinh tuyeit tinh 12'; Đ3. Cau trite dm mot tg ang cau 12Z Chtedng DANG TOAN PHUONG - KHONG GIAN VEC TO OCLIT VA KHONG GIAN VEC TO UNITA 134 A. Tom Vitt 1t thuyeet 134 Đ1. Dang song tuy6n tinh aol xUng va dang town phuong 139 Đ 2. Killing gian vec to gent 135 Đ3. Khong gian vec to Unita 142 B. Vi du 14E C - Bai DM 174 D. Hitting dan hotic ditp so 179 Tai lieu them khan 192 6
  6. Chuang 1 DINH THUG - MA TRAN A - TOM TAT Lt THUYET Đ1. PHEP THE Met song anh o tit tap 11, 2, n} len chinh no duet goi la met phep the bac n, ki hieu la '1 2 3 G G \ I a 2 3 15 del a, = a(1), 02 = a(2), , a„ = a(n). Tap cac phep the bac n yeti phep nhan anh xa lap thanh met nhom, goi la nh6m del xeing bac n, ki hieu S. S6 cac Olen t3 cua nhom S„ bang n! = 1, 2 n. Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich than the cem a n6u s6 - j) (a, - a) am. Phep the a &foe goi la ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6 nghich the ciaa a le. 1 neM s la phep the chan Ki hieu sgna = -1 net} a la phep th6 le va sgna goi IA deu am, phep the a. Neu a vat la hai phOp the cling bac, thi sgn(a = sgn(a) . sgn( ). Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„ • - • , ik doi mot khac nhau dr coo = 12, coo = i3, a(ic) = i1 7
  7. va a(i) = i vdi moi i x i„ ik . Vong )(felt do dttoc ki hieu IA ik). M9i phep th6 dau &tan tfch the thanh tfch nhung yang xfch doe lap. Met vOng xfch do dal 2 dude goi IA met chuygn trf. Vong xfch ••• , ik) phan tfch chive thanh tfch 0 1 , Đ 2. DINH THUG I. Gia sit K IA met trueng (trong cuan sich nay to din yau xet K la &Ong s6thvc K hoac truang s6 phitc C). Ma tran kidu (m, n) vdi cox phan tit troll twang IC la met bang chit nhat gfim m hang, n cet cac phan tit K, i = 1,m, j = 1,n. Tap cac ma tran kidu (m, n) chive kf hieu M(m, n, R). Ma trail vuong cap n IA ma tran co n dong, n cot. Tap cac ma trail vu8ng cap n vdi cac phan tit thuoc truong K ki hiOu IA Mat(n, K). 2. Cho ma tr4n A vuong cap n, A = (ad, i, j = 1, 2, , n. Dinh thitc ciia ma tran A, kf hieu det A la met flan tit dm K dude xac dinh nhu sau: detA = zsgn(a)a mo) E Sn 3. Tinh eh& ceta Binh that a) Neu dgi cho hai dong (hoac hai cot) nao do cim ma tram A, thi dinh auk cim no ddi da:u. b) N6u them veo met dong (hoac met cot) cim ma tran A met to hdp tuygn tinh cim nhUng thing (hoac nhung khac, thi dinh auk khong thay ddi. 8
  8. • c) Ngu mot Bong (hay mot phan tfch thanh tong, thi dinh thitc dU9c phan tfch thanh tong hai dinh thfic, cv th6: f an ail an, de a21 +alci a2„ ‘a n„ a,,, + ani all al; a1,„ all a21 a2 n = det a 21 21 +det —a 1111/ " S ' Ill " S IM / d) Cho A = (Ito) E Mat(n, K), thi = b) a do = aij &toe goi la ma tran chuy6n vi cim A. Ta co detA = detAt. 4. Cdch tinh dinh that a) Cho ma tran A E Mat(n, K). Kf hi'911 Mi; la dinh that cua ma trail alp (n-1) nhan dine bAng cach gach be clOng thU i, cot thu j cut ma tram A vb. Aij = (-1)H M u clucic g9i la pha'n phu dai s6cUa phgn to aii cna ma trait A. Ta cú CAC tong thtic: O ngu i k det A ngu i = k O ngu i x k det A ngu i = k Nhu fly detA = EamAki (k = 1, 2, n) 1=1 heat detA = Z aikAik /=1 9
  9. CUT thac tit throe goi la cang thdc khai trim dinh tilde theo (long hay theo cot. b) Dinh 1ST Laplace Cho ma Iran A = (a,J) c Mat(n, K). Vo; rn6i bQ ;2.••, ix), 1 s i, <12 < . < ik va Oh ik), j2 < ••• <jk n, 1 s k< n, dat A. la 1 =11 < 11-11, dinh thac caa ma tran vuong cap k nam d cae &nig i„ ik va cac cot j, jk cim ma tran A; M. h la dinh thdc cita ma tran vuong cap (n - k) c6 dude bang each gach the clang thu ik va cac cot thu j 1 , , jk caa ma trail A. Ta c6 kgt qua: ik detA = ZED' Si k+31+ A. do j1. jk la k cot cgdinh. Tgng &toe lgy theo tat ca cac bQ ik) sao cho < i2 < < i k Cong thdc troll dude goi la c8ng thdc khai trim dinh thae theo k cot ji, 'along tV, to cú gong thdc khai trim theo k clang. Khi k = 1, to &roc gong tilde da not trong muc a. Đ 3. MA TRANI 1. Ma trgn kigu (m, n) vgi cac phan tU tr8n trang K da dude gidi thigu trong Đ2. Tap the ma tran kigu (m, n) vdi cac phan ti tren tragng K dude ki hiQu la Mat(m, n, K) A E Mat(m, n K) (Woe vigt A = (aii) i = 1, m ; j = 1, 2 n hay ro rang hon: 10
  10. 811 a19 aln aon A= a, 1 a99 amt amt arnn, 2. Cac phep todn tren Mat(m, n, Cho A = (a y ), B= (b,j ) thuOc Mat(m. n, K) Ta cú: a) Ma tran C = (cg) a do cy = aii + &toe goi la tong cua hai ma tran A va B va ki hien la A +B. Ma tran D= (d,,) a do di; = aij - dude goi la hiOu cila ma trail A va B va Id hi'eu la A - B. b) Vdi k E At, ma trail kA c8 cac phAn to la (kaii ) duoc goi la tick cua ma tran A vdi ph&n td k cua trudng K. c) Neu A = (aii ) c Mat(m, n, K) va B = (bp() e Mat(n, p, K) thi ma tran A . B E Mat(m, p, K) ma cac phAn tit &tele xac dinh INN AB = (c,k), a do e ik = Zaijbjk 5=1 &toe goi litich caa hai ma tram B ye. A. Vol A, B e Mat(n, K), to cú det(AB) = detA. detB. 11
  11. d) Tap Mat(n, K) con ma tran yang cap n vdi phep toan cOng lap thanh mot nhom giao hoan, con vdi phep Wan rang ma tran va phep nhan ma trail lap thanh mat vanh khong giao hodn, co don vi. 3. Hang ctia ma tran; Ma trim nghich ddo Gig A E Mat(m, n, K), ta dinh nghia hang ciat ma tran A la cap cao nhgt cua dinh thric con khgc khong rut ra W ma tran A. KM A E Mat(n, K) va hang A = n (ta cling dung ki hi3u hang A la rang A) thi ma tran A goi la khong suy bign, khi do detA * 0 va ton tai duy nhgt ma tran B thuOc M(n, K) A.B = B.A = I„; d do I. lit ma trail don vi. Ma tran B &roc goi 11 ma tran nghich dgo cna ma tran A va ki hi3u la A'. Gig su A= (Au )la ma trail plw hpp cim ma tran A = (ad, Ab la Olga Ow dal see mitt phgn ht aii; At la ma tran chuya'n vi cua A . Khi do: At . detA B- VI DTI Vi cla 1.1. Xac dinh clgu rim cac phep th6 saw a) a 11 2 3 4 5 2 3 5 4 1 r1 2 3 n n+1 n+2 211 2n+1 31 b) 5= I ll 4 7 ai-2 2 5 12
  12. Lai gidi a) Phan tick) a thanh tich cac chuydn tri: 2 3 4 5) = = (1 2 3 5) = (1, 5) (1, 3) (1, 2) (12 3 5 4 1 (chi) 9 la pile)) nhtin cae chuydn tri dude thuc hign tii phai sang trai nhu hap thanh cua the Anh xa). Vay sgna = (-1)s = -1 Co the lam each khan: Cam nghich the cua a la (1, 5), (2, 5), (3, 5), (4, 5), (3, 4). Vay a cú 5 nghich the nen sgna = -1. b) Ta hay tinh sS nghich the cua boa)) vi (1, 4, 7 3n-2, 2, 1 khong tham gia vao nghich the 4 tham gia yen 2 nghich the voi the s6 thing sau no. 7 tham gia van 4 nghich the. 3n - 2 tham gia vao 2(n - 1) nghich the voi the se dung sau no. 2 khong tham gia vao nghich the nao vdi the se dung sau no. 5 tham gia yen 1 nghich the voi the se dung sau n6. S tham gia vao 2 nghich the vdi cae s6 dUng sau n6. 3n - 1 tham gia vim (n - 1) nghich the voi the s6 thing sau no. Cae s6 3, 6, 9 , 3n khong tham gia vao nghich the nao voi the s6 (hang sau 13
  13. Vay co tat ca 2 + 4 + 2(n-1) + 1 + 2 + (n 1) - 3(n -1)n 2 (n-1 )n nghich the trong hoot vi da neu va do d6 sgn S = (-1) 2 Khi n = 4k hoac n = 4k + 1 thi sgn 5 = 1 con neu n = 4k + 2 ho4 n = 4k + 3 thi sgn = -1. Vi du 1.2 _ 1 2 3 Cho phep th'e' f - en dgu la (-1)1 12 3 fn Hay gag dinh da"u dm: a) 1-1 2 b) g = (In fn-) • n ) Lift( gidi: a) Vi sgn f. sgn = sgn (f. = sgn(Id) = 1 non sgn (e 1)= sgn (f) = (-1) k nj b) X4t phep the"a = 1 2 n -1 . 1 thi g = f. a Do gay sgn g = sgnf . sgn a. n(n-1) Nhung sgn a = (-1) 2 nen sgn g = (_])k+C;', 14
  14. Vida 1.3 ChUng minh rang vier nhan mat phep th6 vdi ehuy6n trf j) v6 ben trai Wring throng v6i viac dai cha car s6 i , j a clang drah cna phep the Cling nhu vay, nhan mat phop th6 Nth ehuynn trf (i, j) v6 been phai tunng during voi del eh?, I, j a dong tit 66a phep th6. Lo gidi Gia sif a la phop th6 cho j) la phep chuy6n tri. Xet truong hop nhan ben trai tile la f = (i, j). a. (3 n Gitisi2 a= 1 2 Theo d nh nghia (i, j) = ( 9 2 nj _ (1 2a n ai ) Trunng hop nhan ben phai dude ant Wring fib Vi dy 1.4. Cho f va g la hal phop th6cua n strtn nhien clAu tien. a) Chung minh rang cú the' cilia f va g bang khong qua (n- 1) phop chuyan trf (nghia la ton tai k phop chuyan trf a l , cr2, ak, k 5.11 - 1 g = ak ak_, a,. f). b) Chling minh rang khOng tha giam bat s6 chuOn trf rah trong cau a) titc la en the' chon f va g sao cho khong the dua f vd g bang ft han n - 1 phop chuyan trf. 15
  15. La gicii a) Xet phep the g o f', phan tich g o e' thanh tfch cac vang xich dOc lap T1 , Tp . g o e' = T2 .Ti Neu kf hiOu mi nt do, dai cart yang 'dell Ti thi ± m2 + = rang mOt vong xfch (a l , a2, am) la mOt plop the a cac s6 tv nhien Ui 1 den n sao cho a(a) = a 1+1 (i = m-1) va. a(a„,) = a l, con a(l) = 1 nen 1 yen moi i = 1, .,., m. Vong xfch (al , a2, u„) goi la ce do, dai m. Ta da hiet rad yang xfch do, dai m deu phan tich duo thanh m - 1 chuyen trf. Vi vay g o e' phan tfch duo. thanh 'Lich caa i(mi -1)= n -p = k phep chuyen trf. i=1 Nhungpa. 1 n-1:115n-1 Nhu vay g o f-1 = ak (s, - chuyen tri) TV do g = ak 0 ak.,, 0 0 f, k n - 1 va cac a, la cac phep chuyen trf. i nj b) Cho g = r la phep the ddng nhA 0. 2 ( 1 2 3 on f = . Ta se chUng to rang killing dua Oa 1 2 n-1)n f ve g duo Wang it hdn n - 1 phep chuyen trf. 16
  16. ' WA met phep the" h = 1 ta not rang la 1 2 n phan tit chinh quy ngu i. De" rang neat nhgn vao ben trai cua h met chuygn Lri thi A:C. 1)111in tit chinh quy tang cling lAm la met don vi. That way, ngu ngudc kg, cheng hen i, j la hai phan tit kheng chinh quy cim h ma nob d6i dig hi voi hj ta Jai &tog hai phign tu chinh guy (cum phep th6 m6i) th6 thi: h, vdi moi i = 1, 2, , n-1 thi a co nghich the'. Vi vay, do so' nghich th6 caa a la k < , nen ten tai i o dg oc ii) < ai0+1 ; Xet hoan vi p = 0„) trong do 111 i = a , ngu i # i„, i„ + 1, con p io = a;0+1 , pi+, =a,„ thi HI rang 13 co nhigu hon a met nghich the". Nghia la s6 nghich th6 ciga p la k + 1. 17
  17. Nhan cot thu nhal ciia ma tran A vdi -k rdi cOng vac) cot this k, ta dude: 1 -1 -2 -(n-1) 1 0 -1 - (n - 2) det A = 1 0 0 - (n - 3) 1 0 0 0 Khai trio'n the() dung Ulu n, ta ea: -1 -2 -(n -1) -1 = (-1)" +1. (-1)"-'=1 Cdch 2. Ta tha'y A= B. ca do 1 t1 1 0 B= vi C= 1 11 ma detB =1, detC = 1 nen detA = detB. detC = 1. Vi du 1.9. Hay tinh cosa 1 0 0 0 1 2cosa 1 = 0 1 2cosa 0 0 2cosa 1 0 0 0 1 2cosa 21
  18. Lai gidi: Khai trim dinh thac Lheo cot cub" to co D„ = 2cosa . - 17,1_2 De thay D, = cosa. cosa 1 = 2coi2a - 1 = cos2a. 2cosa, Gia sa D1 = cosia \TM moi = 1, . k. Ta cú Dk,I = 2cosa . Dk - Dk_ i = 2.cosa . coska - cos(k -Da. = (cos(k+Da + cos(k-1)x) - cos(k-1)a = cos(k+1)a. Nhn vay D„ = cosna Vi do 1.10 Hay Dull + a-9 1 0 0 1 e" +e 1 A„ = 1 N x0 1 0 1 eP + e-(1) a do the phan to tren &tang choo chinh bang nhau va band eq) +e-9 ; the phan tit tren hai &tong xien Win nhat \TM (Mang the() chinh bang 1, con the phAn ta khac bang 0. 22
  19. Khai trin theo cot tht nhEt, to c6: An = (e P +e-P)A n_i: e 21' - e-2(P Nlinn xet rang 4 1 = 6 9 +Cc = - 36 -39 e ro e 6 - A 2 = ((i e(P - e(1+1)6 - e -(lrv1),p Girt sit AR - , k - 1, 2, , n - 1. e(-0 -e (P Ta c6 An = (ec e -P)A n _i -A n_, n-1) n4) (:+1)o -(n+1)4' nip - e npe( - e e - e =(eP +e w)e ew e q) - — (n+1)p -(n+1),p e - e Nhu v6}.:. A n = e 1 -e Vi du 1.11 Tinh: l l a 1 a, a n 1 a l: +h i a, a n D = dot 1 a1 a, +b., a n a 1 a.9 a n +b n 23
  20. Lai gidi: LAY ciOng dAu nhan vOi -1 r6i Ong vao cac thing con 1ai , to cú ngay D = b, 1)2 b„. Vi du 1.12 Cho da thric P(x)=x(x+1) (x+n) Hay tinh Binh thdc: P(x) P(x +1) P(x + n) P(x) P(x +1) P(x +n) d = P(n-1) (x) Pth-1) (x+1) Pth-1) (x + n) Pthl (x) Pthl (x +1) P(n) (x +n) gidi: Ta b6 sung de' dude ma Han dip (n+2): P(x) P(x +1) P(x +n) 0 P(x) P4x+1) P4x+n) 0 D = P(n ) (x) Pthd(x +1) P0P(x + n) 0 P(„+l) (x) 1301+1) (x +1) pg+0( x +n ) 1 RO rang det D = d (x +n Nhan dOng 1111 k cua ma trail D vdi dc-ix( 1) k-1 r6i (k-1)! Ong vao clang Hirt nhgt vgi tat ca k=2, n+2). Khi do, phAn tii dung dau cú clang: poc + 0 + P(k) (x +0.(x +11.) k o k = n). k=1 k! 24
  21. n; con phan tit a cot cuoi bang (-1)1T1 (x +n) n+i nghia 13 clang thg nhaa c6 dang: (n+1)! (0, 0, , (+1)"+1(x+n)ni ). (n+1)! Do do Pkx+1.) P(x+n) (x+n)°+i d = del D - (n+1)! PTUx +1) . P(n)(x+n) PT+I kx+1) PT+Ikx+n) Ta ki hieu dinh thge a v6 phai bai C va ma tr5n Wong ring hai (6-1 Vi da thge P(x) = n(x + i) Ken P'"0(x) (n+1) !, vi vay i=0 the s6 hang a dOng cu6i &au bang (n+1) !. dgn gian ki higu va each viek ta dirt xk= x+ k, k = 0, 1, n. d dOng thg hai tit &leg len ciaa ( ta co: (Pw(x0), P("ax,), PT)(x.,)) ((n+1) hco + al , , (n+1) tx„+ a a do al la hang s6 &do do. Khi nhan (long cue"' ding voi l (n+1)! r6i Ong vao clang trail no, ta dtta ma trgn ( ) va dang P(x0) P(x] ) P (xi) ) P(11-1) (x0 ) Pth-e (xl ) PT-1) (xn ) (*) (n+1)!x0 (n+1)!x 1 (n+1)!x n (n+1)! (n+1)! (n+1)! 25
  22. Deng this ba tit dUdi len caa ma Dan (*) co dang (n +1)! 2 (n+1)! 2 xo +ai xo +a,, „ 2 x n +ai xn +a., 2 Cong vac) (long nay hai clang °Ma sau khi nhan voi cac s a, va a1 ta nhan dude clang (n +1)! (n+1)! (n+1)! 2 (n+1)! 2 (n +1) 6 9 xo , 2 2 2 Bang each bidn ct6i nhu vay vdi cac dong con lai, ta clan m Dan ( '61) ve clang sau ma khang thay d6i clinh thfic caa no. c = det = det x on X n n-1 x n1 X0 (n+1)! = (n+vn k=1 IC ! X0 n(n+1) 2 .((n +1)!6'1 (-1) .Dn . J k! k=1 26
  23. 6 do D„ la dinh thiic Vandermonde cua clic so" x„, x„ n-1 D6' thay D„= n(x k - xi )= n (c_i) = 11(n-o!= 1. k>i>0 k>1 1=0 k=1 Vay d = detD = (-1) 2 [(n +1)! in .(x +n)n+1 . Vi du 1.13 Gia sU A e Mat(n, K), A= (au) a do ali = 0 vdi moil= 1, 2 , con aij bang 1 hoac 2001 \Ted i t j. Chung to rang ne"u n chan, thi det A # 0. Lbi gidi: Nhan xet rang neh to them vim mit phan to ajj nao do cha ma tran vuemg A mot s6 than, thi dinh thfic cha ma tran nhan dticic se sai khac vat dinh thiic cha ma tran A mot s6 than. Vi the ne'u to hat di 2000 don vi a nhUng phan t,i bang 2001 cha A, thi tinh than le cha dinh thfic cila A khong thay d6i, nghia la: detA = detB (mod 2) a do B = (14 hi; = 0 n6u i= j va = 1 ngu it j. Ta cú: 0 1 1 0 det B = 1 1 0
  24. nhan Bong ddu veli -1 fee cling vao cac dOng con lei ta dttuc: detB = 1 0 0 -1 Deng vao cot thu nhat ca car cot con lai ta cO: detB = . (n-1). Vey khi n than thi detB la sidle, do vey detA # 0. Vi du 1.14 Tinh Binh theft: ' 1 1 1 1 02 C 3 C I C n+1i D = det q C., C 241 2 Cn+2 c n-1 c n-1 n n+1 C.!;1-11-.) C2-112n-1 > LtlL gill Vi CI, +Clr = C74, nen hieu cUa moi phan tit vol phfin I dUng b8n trai no thi bang phen td thing ngay tren n6. De tir D, ta ldy cot thit n tr3 di del n-1, r6i ldy cot n-1 tie( di cot 2, , ldy cot the] 2 trU di cot thU nhat, ta co: 28
  25. 1 0 0 0 0 C12 1 1 1 1 D = det Cy c?c, C 1 C 1„1 " • cn-2 011 C1,,1 12 C112 2n-2 lai lam nhtt tren, to co 1 0 0 0 C2l- 1 0 0 D = det C3 C13 1 1 n-3 c^-3 2n-4 sau n -1 budc nhit v'ay, to 1 1 0 0 0' 1 0 0 D = det CI3 1 0 = 1. cn-1 Lin-2 cn-3 C I 1 VEty D = 1. 1.71 du 1.15 cos9 - sin9 Cho A= fray tinh An . sin9 cow , Lai gidi: cosy -saw cos9 -sine cos2p -sin29 Ta co A2 sine cost° sing cosy, sin2c cos29 29
  26. = cosky -sinkp vOi k = 1, Gia A k n-1. sink cosk(p Ta ttnh cos(n - lap - sin(n cosy - situp A" = A n-1 .A = sin(n -1)9 cos(n -1)9 2 \ sing cosy , cos (n -1)9 cosy - sin(n -1)9 since -cos(n - lap since- sin(n -1 9 cosy sin(n - lap cosy + cos(n - lap situp cos(n - lap cosy - sin(n -1)9 sirup cosmp - sinmp sinmp cosny = cosny - sinny Vay vdi 11191. n E N. sinny cosny , Vi (11 t 1.16 0 1 Cho A = hay tinh A 200" 0 Lbi gidi: -1 0 0 -1 (1 0 Ta c6 A 2 = 0 -1 1 0 0 1 ma 2000 chia het cho 4 \ray A200° =(A)500 =I, (I, la ma tran thin vi cep hai). Vi du 1.17 Ma Iran vuong A e Mat(n, K), A = (ad throe goi la Ina tran phan del xang netu aii + = 0 vdi moi i, j = 1, n. 30
  27. Hay chang minh: 'rich caa hai ma tran phan doi xang A va la mat ma tran phan doi 'ding khi va chi khi AB = -BA. Lai gidi: Gia sid A = (ad, B = (bid d do ait + ai, = 0, b y = 0 di mm L j = 1, , n. Dat C = A. B = (c,); = =1 D = B. A = (c1,0 = Ib ijajk id] Ta ca.: c,k = Za ji b ik - Eb kiaji - d ki Nha vay AB phan del xang c, = V i , k. tt=> = vet mot i, k c> AB = -BA. Vi chi 1.18 Cho A, B la hai ma tran vuong c9p n. Hay minh let(A.B)= detA. detB. gidi: G1a. sit A = (a,,) , B = (It o ) yea i, j = 1. Ta lap ma tram an a1.,a l n 0 0 a91 a22 a2n 0 0 ant a n11 0 C= - 1 0 bth 0 b21 b2n 0 0 31
  28. Khai tri6n theo n clang dal) (thee dinh 12) Laplace), ta co detC = detA. detB. (1) Mat khac, bi6n &it ma Iran C bai phep bi6n ct6i sd ca) sau: Nhan cot thu nhal vat b 1, cot thil hat Null cot thin n vet btl; r6i Ong vac) Ot thu n + j (j = 1, 2, n), ta dude ma tran D Bang sau ma dinh thiic cua D va cua C bang nhau: al a a d ta d a21 am am ann dm d , , d nn D= -1 0 0 0 -1 0 0 a do = iar ikbki , nghia la (dij) = A. B. k=1 Khai tri6n theo n cot cuoi (theo dinh 157 Laplace) ta cú detD = det(dij) = det(A. B) (2) Tit (1) va (2) va do detC = detD nen ta co: det(A0 B) = detA . detB. Vi rtu 1.19 Cho X la ma tran vueng cap n. Chung minh rang X giao hean vOi moi ma tran vueng Ong ca) = X co clang XI,,, a do I„ la ma trail dun vi cap n. 32
  29. Lari gidi: Ni111u X = 1 thi re rang X giao ho an vdi moi ma Han \along cling cap. Ngnoc lai, gia sit X = (X„) giao haul vdi moi ma tran yang cdp n. VOi i„ # j 0, to chiing minh x inio = 0. Mudn stay chon A = (ad trong do a joio =1 con one phan tit khac ddu bAng Thong. Phan to ding io cot j„ cim ma tran XA Wing x. •' can phan tit a thing io cot j, cim AX la 0. Tir di6u kien AX = XA suy ra =O. Nhu y X co dang: k r 0 X= „ 0 k„ Cho ma tran A = (a) a do a], = 1 vii moi i, j. Khi do phan tit o dung i cat j cim ma Han XA la X , con phan to 0 ding i cot j dia ma tran AX la 2. nen k , = Vi vay: X= 7. I,,. Vi du 1.20 Cho ma Iran cap n: a b b a A= b a) Chung minh detA = (a-b)° (a + (n-1)b). b) Trong trudng hOp detA s 0. Hay Huh ma trdn nghich dao A+ oda ma tran A. 33
  30. Lai giai: a) Gang the clang vao dOng this nhat ro'i nit a + (n-1)b a clang dau, ta (Woe 1 1 1 b a b detA = (a + (n-1)b) x b b a Lay dOng chin aria (Anil th.fie tit nhan voi -b r6i Ong vao the dong sau, ta eú: 1 0 0 0 a - b 0 detA = (a + (n-1)b) x 0 0 a - b detA = (a + (n-1)b) . (a-b)" -i. h) A kha nghieh detA 0 .(=> a b va a + (n-Gb O. Goi B = (bii) la ma tran nghich (lac) cila A = (a„). 1 Ta Wet rang bid = detA . A„ a do A.; la phAn Phu dai so etia phan trong ma tran A. a b b a Vol mdi i thi = la nh thiic cap (n-1). b b a Theo phan a) thi A i = (a-b)"-2 . (a + (n-2)b). 34
  31. A a+(n-2)1( Vay b„ - det A (a 1- (n -1)1).(a - b) Vol i # j = (-1)H , d do kl„ la chub thue e6p n-1, co dupe ba' lig each x6a &Ong thu i va cat tha j eila ma tran A. Do A dovi ming nen = Gia sU rang i < j, khi do cot thfi i va dung thu j-1 cua Mo gain town nhang phan t& b. N6u d6i eh6. Bong j - 1 len tren Ming dau (Mu nguyen cac dung khac), rei lai MS( nit i len 6;4 thu nhai (va van gib nguyen the cat khac), thi to (Moe: m = = (-1)'34 b.(a-b)"-2 . -b Nha v(6y b dot A (a +(n -1)b)(a - b) Do do a +(n -2)b -b 1 (a +(n -1)b)(a -b) -11 a +(n -2)b C BAI TAP 1.1. Chung to rang mai phep chuyan tri dm (n 2 2) lh mot phep the le. 1.2. Hay phan Lich eau pile') the! sau thanh tieh ene phep chuy6n tri 35
  32. 1 2 3 4 5 6 7 8 ' a) 8 1 3 6 7 5 4 2 ) ( 1 2 3 4 5 6 b) 6 5 1 2 4 3 1.3 Tim s6 tat ca the phep the a e S„ sao oho a(i) x i vol mm i = 1, 2, n. Cluing t8 rang khi n chan, so" one phep th6 clang tren la m54 3616. 1.4. Ki hi8u (n, k) la secac hoan vj rim 1, 2 n ce dung k nghich the Chiing minh cong thile truy 146i sau: (n+1, k) = (n, k) + (n. k-1) + + (n, k-n). vdi guy vac (n, j) = 0 nevu j 9 1.5. Ta goi 45 giam ena phdp the f Ia hi5u cna s6 the phan tic khting bat dog (nghia la s6 the pilau tit i ma f(i) i) va s6 clic Yong xich 2 va a la so? nguyen, nguytin to' d61 vdi n, thi tthing Ung ki—>r(ak, n) la mot phan tit cua S„,, k e 11, , n-1}. 1.7. Cluing minh rang moi phep th6 cap k > 1, dou phan tich dupe thanh tich nhiing chuydn tri dang (1, i) vdi i = 1, k. 36
  33. 1.8. Xac dinh davu cua cac phop the' sau: 2 n n +1 n + 2 2n 2n +1 3n a) I 1 43 6 3n 2 5 1 3n-2 1 2 n+1 n+2 2n b) 2 1 2n 1 2n-1, 1.9. Chgng mink rang: a) Dinh thiec cap ha ma cac Phan to Wang +1 hoac -1 le mot s6 b) Dinh attic ay Ion nhig la 1. c) Dinh thew c)41) ba ma cac pga'n tit la 1 hoac 0 dot gia tri IOn nheit biing 2. 1.10. Tinh dinh thew cap 2n D = det(cM), a vai i= j dO di; = b yen i + j= 2n +1 0 i=j vet i+j 42n+1 1<i, j S 2n. 1.11. Cho A = (an) la ma trn cap n, a1 e R. Chung minh rang detA khong thay dei, neit cac phein tei a i; ma i +j 15 dooc thay bai so' dal cga no. 1.12. Chung to cac dinh thew sau day bang khong: 1 cosa costa cos3a cosa costa cos3a cos4a a) costa cos3a cos4a cos5a costa cos4a cos5a cos6a
  34. -b1 a1 - b2 al -bn a, -b1 a2-b2 a, -b a h) ; n > 2. an -b1 a n -b.,-bn n 1.13. Cho hai ma tran A. B sao cho: 5 11 1 14 A B = I, B. A - J1 25 ) 4 y Hay tam x, y va A, B. 1.14. Hay khai tridn dinh thiic va chilng minh: 1 1 2 c+- b+- c b 1 1 a) c+- 2 a + — (abc -1) c a b + 1 a+ 1 2 b a (a+b)2 b 2 a2 b) b 2 (b+c) 29c- = 2(ab be + ca)2. a = c= (a + c)2 1.15. Way tinh cox: dinh thric cap n saw 1 1 1 0 1 1 1 -1 0 1 -1 0 1 a) Dn = -1 -1 0 b) = -1 -1 0 -1 -1 -1 38
  35. 1.16. Hay tinh dinh thdc sau: 1+x'2 x 0 0 x 1+x 2 x 0 a) D„= 0 x 1+x 2 0 0 1+x 2 nghla Dn la dinh thiic cap n ma cac phan tit tran during cheo chinh bang 1 + x2, the plidn tit thuOc hai [Wang chat) grin during cheo chinh bang x, the phOn tit con lai bang 0. 2 1 0 0 1 2 1 0 0 1 2 0 b) 1 0 0 0 1 2 1.17. Cho da thite P x) = (x - al)(x - (x - a„). a d6 a, la cac so thvc dot mot phan biet. Hay tinh climb thde sau: P(x) P(x) P(x) x-a 1 x-a, x-a n 1 1 1 a l a, a n a; a n-2 an-2 an -2 1.18. Xet hai ma tram ph& cap ha: 'a b c ' 1 1 1 A= c a b va J= 1 j b c a, v 1 j- 2n . 2n 1 dO j = e = cos— +. sm- = . 3 3 2 2 39
  36. Way chang minh det J # 0. Hay tinh ma tr'nn AJ. tit do suv ra gin tri detA. Hay nen hal Loan Wong to kin A ya J cite ma 1.rn cap n. 1.19. a) Hay tinh dinh thing cap n san: a+13 (A.13 0 0 0 1 a+11 a.{3 0 0 14„ = 0 1 a+ ^ 0 0 0 0 0 1 a+ 0 1)) Chung to rang dinh tithe sal' khong plat thuhe vao y,, ) ••• Ynt 1 1 g1+ Yl x 2 + y1 (x n +y 1 ) (x 1 1111 1)(x1 11 379) (x9 + N.1)(x9 + y2) (x„ +y 1 )(x 1 +y2) D„= n-1 n-I 11 (X1 + Yi ) Fl(x.) +yi ) n(xn +)'i) 1=1 i=1 1.20. Cho ma 14.'0 ( 1 -2 1 \ A= -1 1 0 -2 0 1 Hay tinh A u1°. 40
  37. 1.21. (ha si:t ma trgn A e Mat(m, n, va rang A = 1. Jhisng minh rang cac ma trgn B e M(m, 1, K) va C e Mat(1, 1g A= B. C. 1.22. Chung minh rang ma trap A =( a bd ) thOa man pIntong trinh: X2 - (a + d)X + (ad - bc) = 0, 0 do 12 la ma Iran don vi cap hai. 1.23. Cho ma trgn 9. 1 0 v 4 1 A= vOi 4 x 0. Hay tinh A-'. 90 1.24. Cho ma tran vuong c5p 4. cosa sincx cosa sina cns2a :;in2a 2cos2a 2sin2a cos3o. sin3a 3cos3a 3sin3a cos4a sin1a 4cos4a 4sin4a Chung minh rang A khg nghich khi va chi khi a ♦ kg (Ice Z). 1.25. Cho ma trgn A = (a) e Mat (n, 1H) ma the phan tit doge cho bai tong that: (-1)H- CV yin vOi i = j 0 vai i>j do Chi.,' - . Chung minh rang A2 =1,1. 6 kHrn-k)! (9, la ma Iran chin vi). 91
  38. 1.26. Gth sa X = ())) e Mat(n, R), 1-1 a do xii 4- (-1)")! Chitng minh X' = I. 01- 4, Chit Se voi aeR k e N, ki hiqu a(a - 1) (a - k +1) ())) ) 0 ; - )- 1 )a, k! a 0 1.27. Gia = yXx„ x.„„ x„) vdi (i = 1, 2, , n) la ham cua cac bi8n doe lap x 1 , x2, x„. Ma tran J = J(Y, X) = [ aYi dx•1 i dliciC goi ]a ma tran Jacobi cua phdp bi6n din, con Binh thud dm no duck goi la Jacobien dm phep biers den do. Bay gin /cot m6i quan hq gilla n2 ham vii va n 2 bien xii dune cho Isdi ding thiic: Y= A.X.B, a do Y= (yi) , X = (xi), A . B e Mat(n, R) la hai ma tran the trn6c. Chung minh rAng det(J(Y, X)) = (detA)" . (detB)". 1.28. Cho X = (x„) e Mat(n, R) la ma tran tam gide dudi; va Y = X. X. Chung minh rAng det(J(Y, X)) = 1.29. Cho Z lA tap cac sqinguyen; A, S hai ma tran vu8ng cap n, cac phAn t> la nhilng s6nguyen (ta vie) A, S e Mat(n, Z)). Hun nua detA = 1, det S x 0. Dal B = . A. S; Chung minh rAng cOs6m nguyen dding de Bm e Mat(n, Z). 42
  39. 1.30. GM sit trong ma tran A = (a u) c Mat(n, R) da cho rude tat ea the phan ti a d (i # j). Chang minh rang cú thk dien Mo &rang char) chinh the s60 hoc 1 de ma tran A kheng suy Bien. 1.31. Tim tat ca cac ma tran A e Mat(n, K), A= (dj), 0 na tan toi ma tran nghich dao A-' cling cú the phiin t5 khOng am. 1.32. Cho ma tran vuong A co the phan to la s6 nguyen. rim diet' kien can va du de ma train nghich dao cung c6 the -Men to la s6 nguyen. D - HDONG DAN HOAC DAP S6 Xet T e S„ (n 2 2) gie sii i < j va T = j, T (j) = T (k) = k vdi moi k s i, j. Khi do the nglach the eaa ji, k} Nob < k < j j/, j} Nob / i + 1, j - 1 With vey co tat ek la U - i) + (j - i - 1) = 2U - I) - 1 nghich the Vi so nghich the le nen t la phop the le. 1.2. a) Phop th6 da cho phan Deb thanh hai yang xich chic lap (1 8 2) (4 6 5 7) = (1, 2) . (1, 8) (4, 7) (4, 5) (4, 6). b) (1, 6, 3) (2 5 4) -= (1, 3) (1, 6) (2, 4) (2, 5). 1.3. DS. S6 Dat ca cac phep the a e a(i) # i vai moi f n i =1,n la n! E (- 0 k0= kl 1.4. Xet tap A gam at ce eke Minn vj (Ma 1, 2, , 14, n+1 co thing k nghich the. Ki hieu A i la b° phkn eaa A gdm cac imam NO ma n +1 . NMI yky A, (i =1, 2, , n-F1) a n , a n+1 ) ai = la nhang tap con rot nhau caa A va A = A, U A2 U U U AnA. 43
  40. Xet Anki = {a = (a p a 2 , , an+1)1 = n +1} NMI 0y so cac nghich the cna a bAng se cac nghich the cue 1 9 nghia la bling 1+ Dieu do cheng to so ea( U Ch 2 Jhfin to cria A„,, bang (n, k). Xet tap Ai (i = 1, , n). cis su a e Ai, a = (a, aj, a„4 ,) Theo dinh nghia a ; = n+1. Nhti vay (aj, khong la nghich th( vdi j i. Do do a, tham gia vac n+1 -i nghich the Xet hoan vi a' = (a l .— aa,_„ (>61, cCia S2 S5 nghich the maa a' bang sernghich the cna a trii NM; vay ta cú met song anh tit A; len tap cac hofin vi cua S„ co thing k-n-l+i nghich the., do do so' phan t& cera A; IA (n, k-n-1+i). Td do, sephan to cUa A la: (n + 1, k) = (n, k) + (n, k-1) + (n, k-2)+ + (n, k-n). 1.5. a) Xet phan tich f = a l o a, o a,„ thanh tich cac veng xich dec lap di) dai > 1. Gia sa do dai al, la dk; ta they f(i) # i khi va chi khi i thuoc met trong cac yang xich do. Vay do giam coal IA d,+€1.2 M6i vOng xich dai dk phan tich dude thanh dk-1 chuydn trf. Do vay f phan tick deck thhnh d 1+ + (.16-1n chuyen trf. Do do kha'ng dinh a) driec chting mink. b) Goi / a di) giam can. f. Theo a) f phan tich (kw thanh / chuydn trf. N5u cú phfin tich f thanh h chuyen tri nao do, ta phai chring minh h 2 1. Ta cú bd de sau: Neu a, 13 tham gia van met yang ;rich cas phep the f, thi khi nhAn f vdi chuydn trf (a, f3) (ve ben trai hay ben phai) yang xich dji not phan thimh hai \Tong xich d'ec lap. Con neu a, tham gia vao hai vOng xich cua 44
  41. Thep the f, thi khi nhan viii chuyen tri (a, (i), hai yang xich se Thep lai lam met (136 de nay a thing chfing minh). Tr/ do neu g a phop the. va T la met chuyen tri tin dp Mem cua g o T khong vire); qua do giam cfia g ceng them 1. Vi the nen f phAn tich luec thanh h chuyen tri thi do giam cUa f khong \wet qua h. 1.6. H.D. Do a va n nguyen t6 vfii nhau, nen a k khong chic het cho n vdi mm k = 1, 2 n-1. do de r(ak, n) la nhfing so phan hied. 1.7. Vi moi phep the phan tich due() thanh tich cat ghee chuyOn tri, nen chi can chfing minh bai town cho phep chuyen tri(1,j)e Sk vei 1, j # 1. Ta ce (i. j) = (1, i) . (1, ) . (1, i). 1.8. Xem vi du 1.1 - 1-1(n+D a) DS: (-1) 2 . r(1121) h) DS: (-1) 2 ykk =(a 2 _ 1.10. Iasi trien cot dau, to ca. D " " 1 2n-2 • Do D2 = -13.2 nen D 2n = (a2 1.11. A = (aid , detA = E sgna ai,(1) ana(N E S Trong moi tich a ko(k) angfrik , tong E(k +cr(k) )= n(n +1) k=1 la sir than; nen re met se cliSn the thfia so ak.,0.;) ma tong k+G(k) le. VI Vey khi thay ha; nen i + j 1e, thi cfic tich anc(k) khong dei, do do detA khOng del 45
  42. 1.12. a) COng dOng thu nhat vdi (long the./ ba, ta dude don tY le vdi (long thu hai. b) Nhan dong thd nhat vdi (-1), rdi cUng vao die (long th hai va Ulf( ba, ta dupe hai dOng CY 15. 1.13. Ma tran A, B e Mat(2, K). Ta cú hai bat bian detAB detBA va tr(AB) = tr(BA). Vi vay: fx+y=30 {x = 20 {x =10 hoar x.y = 200 {y=10 {y=20 20 14 \ a) x = 20 va y = 10 o BA= 14 10 20 14' 5 11 Ta co ABA = A A. 14 10 , 11 25 2 5k -14X Tit do ta tinh dude A = , A E R -11k 0 5 11 va B = A - 11 25, 10 14 b) x = 10 va y = 20 BA= :14 20, 10 14 5 11 -3 5 l ABA = A X A= P. 14 20 4 11 25, X 55 32 1.15. a) Di, = 1. b) An = -A„. 1 + 4,1 = 1. Tit do suy ra A2p = 1, A2p+, = 0 46
  43. 1.16. a) Khai tri4n D„ theo cot thu nhat, to cee DT, = (1+ x 2 )Dn_i -,4211)„,, , D, = 1 + x2, D2 = (1 + x') 2 - x' = 1 + x1 + x4 . Taco: Da -D„-1= x2 (D„_i -Dn _2 )= x 1 (13 1-2 - D n_3 ) x 201-2) (D9 _ D1). x 21 Tct do Dn = Dn _1 + x 2” Do vay Dr, =1+ x 2 + x 3 + + x 2" b) Ap clung eau a) vdi x = 1. DS (n+1). 1.17. Khai tri on. A(x) then (long tha nhat, to co: P(x) P(x) A(x) = D(a 9 , an ) x - a l - I3(x) + ( 1)"7 ; a 46 D(a2, x -a n la dInh Dane Vandermoncle eaa cac s6 a2 , , an; Cho x = a,, to co A(a l) = (a, - a2) (a, - a„) n(a j>i22 A(a,) = (-1)" n(ai -a1 ). j>141 Thong ta A(a2) = = A(a„) = A(a 1). Da tithe A(x) bac nhO hOn hoar bang (n-1), co the gia Da hang nhau tai n diem phan biat, vi vay A(x) la hIing. Tit do: A(x) = (-1) 11' xFr (a 1 -a.). ISIS“ 47
  44. 1.18. J In ma tran Vandormonde vat (Me s6 biet nen dot J * 0. Ta a+b+c a+ bj+ cj2 a + bj 2 + ej A . J = a+b+c c+aj+bj 2 c+aj2 +bj • a+b+c b+cj+aj2 b+cj2 +ttj dotAJ = (a + b + c) (a + bj + cj 2) (a + bj 2+ cj)det 1.19. a) Khai trien theo cot the] ula. La D„ = (a + a0Dô-s. Ta co: D, = a + 13, 1/2 = az + an + 13 2, k Gia sit D k = Ea i f3 k-i (k = 1, 2, n-1). =o n-1 -1-i Ta co = (a +p) a i r - an Ia'0 2-2-I i=o i=o n-1 n-1 n-2 = a i+lpn-i- orkira .2 a al n n-i-1 1.0 i=0 i=0 n-1 n-1 = r3 n ct i v_i - i=1 ]=1 = E =o Mtn tray D„ = Ear . i=o 4 8
  45. b) Cheng minh qui nap thee n. 1 1 I, n = 2 o D2 = = x2 - x, kheng phi thuec y,. + y i x, +yi n = 3 D„= (x, - x,)(x,, - xi)(119 - x1)- Gia stir Dk = 11(Xj -x i ) vdi moi k = 2, n-1. i Si k • (-D kjjl 11(x2 +y 1 ) (x/ +y 1 ) lxk ixk Net 14(y 1 ) k=1 n _ x, n (xk k=1 (xj -xk ) jxk P(y,) 13 da thvc bUc < n - 1 doi vdi y,. Vbi y, = -x; win moi i = 1, 2, Ta co P(-x,) = 1, do de P(y,) = 1. Tiido ding thirc quy nap three chiing minh . 49
  46. 1.20. agA=B+I3 L la ma tran don vi, 0 - 2 1' 0 0 0 \ B= -1 0 0 , t3do B2 = 0 2 -1 ; W = 0 -2 0 0 \ 0 4 -2 Ding khai trie'n nhi thric Niu-ton to co: A100 = (0 , 13 ).100 100;99 13 +100.B+ u 1- 2 1 -200 100 Suy ra A 100 = -100 9901 -4950 -200 19800 -9899 1.21. Vi rang A = 1, nen tat ca car (long tS, le vb7 mot dong. Wiy ma tran C E Mat(1, n, K) la ma trqn dong do. Giq sv clang thd icim ma trqn A bang b,. C, lay B = (b) la ma Lean cqt. Ta co A = BC. 1.23. A-1 = 1.24. Tinh detA = Tit db suy ra ket qua. 1.25. Bat A2 = (b (), khi bi; = iaik a k; k=1 + Vdi i > j, 13;; = 0 vi = 0 ne'u k j con k > j thi ak; = O. 50
  47. + VOi i a. flat X- =(3.7 .5 ) th y], = [_,x lk x kj k=1 • •• = = (do dp dung nhan k=r n-k, 3 - i , xot tren). 51
  48. Gia sit = (20, to ed za = Zyikxkj kst in-i‘ - kbl Ora k=1 Tr -1, „n - j, „ iEnn+1+1+k n- i k ki=1 „k -1 n Neui j z,i = E Enh-Ft aXt<l) t b day t = k -1. a = n - b = n - j). Nha vSy zij = D_ ob+t t! b! a<t<b a!(t-a)! t!(b-t)! ( _ i ) 1)-0 b b ! x a 1)+1 t-a Cb 1) C b_ik a<t<b (b - t)! (t -a)! t=a Ixa = (-1)am s=0 Nha vky = 8,j nen Xx = I„. 1.27. Trade het to Chang mink rang ne!u Z = AX, a (Id Z, A, X e Mat(n, R.) thi det(J(Z, X)) = (detAr That vay, gia sit Z= (20), 21; = aik xkj . k=1 52
  49. aik n6u / = j Tif do (1) 0 nau / j • Ta coi ma tran J(Z, X) E Mat(n2. 1K) vdi can dOng va can cot duct danh s6bai can cap c6 thit Lu (i, j) , 1 1 X = (xi) e Mat(n, K); x,, = 0 vdi i < j. Ta cú Y = X. Xt = (Y,) Y;i = xi (x k Exik jk k=1 Nhv vay Yii = 53
  50. x is ngu i=j,r=j ay'J ngu r = = j, j Ta cú (*) 2/c4 i„, neat r = i = j 0 trong the twang hop con loi. Ma trOn (1(Y, X) e Mat(n 2, R), the (Ping nọ cac cot dttoc darth s6 nhu trong bai 1.27. Phan tn a hang (i, j), cot (r, s) Pt 13 o rs Ta nhan tha'y vdi (i, j) < (r, s) (nghia Itt i < r ngu j = s hotic dy j < s) thi - 0. Mt n; Nhu vay J(Y, X) la ma tran tam gitic &RR OY • 2X 11vat i = j Xet 13 . Do (*) to cú ax ,j aX X JJ j aXij _ Nhu \ray det(J(Y, X)) = - 'J=1 jX j1 - n(2xn)=- =1l fl JJ j1 J 1.29. Goi A = (AO la ma tram phu hOp cat) A. Ta co A41 = 1 At - A l e Mat(n, Z). det A Ta co B = SI1 . A. S Brn = S4 . S VI ma tran S g6m cac so nguyen nen k = Ida S e N* (do gia thigt dotS = 0). Vol m6i se; nguyen dming p, )(et ma tran AP e Mat(n, Z), gia sit A P (ap ; goi R la set du khi chia a!; cho 54
  51. k = Idet S. Idat = 0'0 vi 0 q sao cho the ph&n tii taring ung cad AP va A P bang nhau thee modk. Do do AP = +(detS).0 , C e Mat(n, Z). Tit do: = +(detS).C.(Aql . that m = p - q. Rhi do: I - 1 A"'S=1„ +detS.S -1 .C.A-4 S; Vi S e Mat(n, Z), (detS). Sl' = S i la ma Iran phu hop cua S, nen S e Mat(n, Z) va vi vity: B" c Mat(n, Z). 1.30. Ta chfing minh quy nap then n. n = 1 : hien nhion, all alp n=2, A= =an a22 -a 21 at2 . a21 22 Neh cho track a2 , . a1 0 t 0, to chon a, 1 = a22 = 0. Con nein a21 . a u, = 0, La chon a„ = a 22 = 1. Nha vay n = 1, 2 thi bad Wan dung. Chi sit bai toan dung laid moi dinh tithc cap < n - 1. Ta chiing minh no dung ved dinh at& cap n. Gia silt A = (ap) e Mat(n Xet A„ la phan phu dai so' caa phdn. tit an . Theo gia thi6t quy nap, to da Hein dude a22, a„„ de? A, 1 0. 55
  52. DAt all = x, khai trien theo clang thu nhAt ta co: detA = x. A, 1 + ZaliAli Ta co I:A uAli la hang set NAu hang so nay kluic khang, ta J=2 chon x = a„ = 0; nAu hang s6 do bang killing. ta chon x = a t , = 1 se. clime detA 0. 1.31. Trade he'd ta chting t3 rang nen ma Iran A va A-' E Mat(n, co the phAn t& kheing Am. Chi a m61 cot cila A co dung met plan ThAt vAy, gia sa a cot thu j cua A co 2 secdim/1g, a clang i, vA dOng i2. Chon clang k vdi k # j cila ma tran A'. VI A-1A = I„ non tich cua dOng k cua A' vdi cot j CEla A bang 0. Nhu va t), cac phzin t0 cfm cot thil i, va i2 nam tren clang k deli bang ki:Mg (k x j). Do do hai cot i2 va i2 cua A-1 tY le vdi nhau. Diu do clan deIn mau thuan. NMI vAy m61 cot cim ma tram A co dung met s6 clueing (con lai dgu bang 0). Dicing te, m6i deng cila ma tran A clang cú dung met s6 throng. Giao hoan cac demg (hoac cac get) ta duoc ma tran chg.°. Nguoc lai, tat ca the ma tran 'Then dude tt ma tran cheo (ali > 0) bang each chuygn clang host cot deu kha nghich. Ma trAn A-I nhAn dude to ma tran A bang each Iay nghich dao cac phan to khac kitting rdi chuyAn vi. 1.32. DS det(A) = ± 1. 56
  53. Chuang 2 KHONG GIAN VECTO - ANH XA TUYE-N TINH - HE PHUONG TRINH TUYEN TINH A - TOM TAT Lt THUlt Đ1 KHONG WAN VECTCS 1. Dinh nghin: Gin sit K la mot traong. Mt tap hop V kirk rung cimg vdi hai phep town "+" : V x V —> V (a , p) 1—* + va phep than " •" : K x V —> V (k, a) 1 > k. a dime goi la mot kheng gian vec to tren twang K n6u no thOa man cac tinh chgt sau. mot k, /, c K va moi a, p, y, c V, to CO: a) a +p=p+ a b) (a+ p)+ y= a+(p+ y) c) CO phlin tit 0 e V sao cho: a + 0 = 0 + a = a vin moi a c V. d) Vdi a e V, ton tai (-a) e V sao cho: a + (-a) = 0 e) k(a + is) = ka +103. g) (k + /)a = ka + la. h) (k. / )a = It (/ a) k) 1. a = a, 1 la phan ta don vi cita truang K. 57
  54. Mot kh8ng gian vec CO tren throng 1K con &foe goi K - kKong gian \ecto. Khi 1K = R, kh8ng gian vac to &loc goi la kh8ng gian vec to thee. Khi K = C, kh8ng gian vac to dude goi la kh8ng gian vecto phiie. 2 - Sit crOc 15p tuygn tinh vit phu thqe tuye'n tinh Vec to 13= k,a, + + k„,a„„ (a, c V, k e K) goi Ht met to' hop tuyo'n tinh cua cac vec to 4 0 = 1, Ta ding not p bieu thi tuy6n tinh qua cac vac to a t, , an„ MOt he vac to la,, •, caa V dude goi la he phu thmic tuyeIn tinh nen c6 cac s6 = 1, m) kh8ng deng thoi bang kheng cna ]K sad cho = 0 . Met Oat hkeu along &king: 1=1 he (st„ am) ka he pha thuec tuy6n tinh nen c6 met vec Lo nac de cim he bleu thi tuyen tinh qua cac vec to can lai oda he. Met he cac vec to kh8ng phu thueic tuy6n tinh doge goi he doe lap tuyen tinh. Nhu \ray, he {a„ am} la he dOc lar tuyen tinh ned.1 moi t6 hop tuy6n tinh k iai =0 to suy ra k, = = km = 0. Cho he 14„ aid cac vec to doe lap tuygn tinh caa kh8ng gian vec to V ma m6i vec to caa no la t6 hop tuyeIn tinh caa ca( vec td cim he ([3„ thi k < 1. . Cho he vec to {di } ; e trong kheng gian vac to V, I la tap chi so gfim huu han phan tit He con (a3) jedcI goi la he con de( 58
  55. lap tuyen tinh t6i dai cim he da cho neu n6 la he dec lap tuyen tinh, va nen them bet kST vec td ak nao (k E I \J) thi ta dude met he phu thuec tuyen tinh. Cho he hau han vec td {a, , , am} trong khong gian vec td V, thi s6 phain tit cim moi he con dee lap tuyen tinh t6i dal ciia he tren deal bang nhau, so/ do duoc goi la hang cim he vec td {ao , 3 - Cd sd va so clueu cua khong gian vec td Met he {e„ , en} the vec td doe lap tuyen tinh am killing gian vec td V duos goi la met co sa oda V nen mm vec to cim V deli la t8 hop tuygn tinh cim vec to {c„ , e n}. Khi V c6 ed sa g6m n vec td thi moi co sa cim V deu c6 dung n vac td. S6 n goi la so" chigu cim V, Id hieu dimV. Neu kleing tan tai mat cd sa Om him han vec to, thi V goi la khong gian vec td v8 han chieu. Cho co sa e = le i, , ej, yen vec tO bet kS7 a e V, ta co a , , x„) dupe goi la cac toa doo cim vec td a del vOi ed sa {e„ , en}, ; la toa de thu i cim a doa vdi cd 56 do. Gia sit co ed sa khac c = {c,, , E„} ciaa V, ma si = ]=1 = 1, n). Nen vec td a cú toa di) (x i) trong cd s6 va c6 toe de' (x11) trong ed sa thi ta c6: = Ec oxli , , n. Neu ta ki hieu ma tran C = va ma trail X = (x), X' = (x') la cac ma tran cot, thi X = C X'. 59
  56. 4 - Klaang gian vec tei con va klaing gian vec td thacing Tap con khong r6ng W ciaa khong gian vec to V duoc goi IA khong gian vec to con ciaa V nen W la khOng gian vec to, voi cac phep toan ciaa V han dig tren W. Tap con khong rang W dia. V la khong gian vec to con ciaa V khi va chi khi W on climb dna vdi hai phep toan dm V, nghia la vdi a, 13 E W va k E K, thi +p E W va k a E W. Cho W, WEI la cac khong gian vec to con cria khong gian 11 vec to V, khi do nW, IA mOt, khong gian vec td con Gem V, IA khong gian con ldn nhat nam trong moi W i, i = 1, 2, , n,,. Cho tap hop X c V, khong gian vec la con be nhat cna V chga X duck goi 1a bao tuygn tinh cila tap hop X, ki MO hay Vect(X). Neu X = , bao tuygn tinh ciaa X thick ki hieu la . Cho W„ , ho nhUng khong gian con ciaa V. Khi de bao tuygn tinh ciao, tap hop W,U UW E, dude goi la tang cim cac khong gian con W, ,W„, ki hien Ia W, + + W„ hay ZW; . Ta thay rang a e EW; khi va chi khi a = Za i , a, e i=1 Neu moi e /113/4 , a vigt chicle mat each duy nhat a i=1 dang a = a, + + a„ , a, e thi t8ng W, ch.toe goi la tang I=t true tigp cua n khong gian vec td con (i = 1, 2, , n) va ki hieu la W, e W2 ED đW„ hay S W,. =1 60
  57. Gia sit V la klafing gian hitu han chikt,W, va W2 la hai khfing gian vec to con dm V, khi do: dim(W,+ W2) = dimW, + dimW2 - dim(W, fl W7). Cho W la khong gian vac to con cim khong gian vec to V. Ket quan he Wong throng tran V: a-Paa-Pe W. Lop Wong :lining cam vec to a dirge ki hiau la [a]. Tap thudng V/W vdi hai [top toan: [a] + [in = [a + [I] va k[cx]= [kal vdi moi a, 13 E V va k e K, lam thanh mat klihng gian vac td, durie got la kitting gian vec to thtfong (ciia V chia cho W): dimV = n, dimW = in (0 m n), thi dim V/W = n - m. Anh xa it : V -> V/W ma a(a) = [al (Inge goi la phdp chi6u tdc. Đ2. ANH XA TUYEN TINH 1. Dinh nghia. Cho V va W lh cac khong gian vec to tram twang K; Anh xa f: V -> W duo° goi la anh xa tuye"n tinh (hay itIng ca'u tuy6n tinh, hay toan tit tuyan tinh) n6u no bao phop toan cua khong gian vec to, cu the' la: veil moi a. p e V. 1119i k e IK, to ce: P) = f(a) f(P) f(k a) = k . f(a). Anh xa tuy6n tinh dude goi la don din nett ne la don anh, :oan can n6u n6 la than anh, va dang 6.'11 n6u n6 la song anh. 61
  58. Hai khong gian vac td V va W chive goi la clang cdu vol nhau nAu ce met ding cau f tla V len W. 2 - Cac phep than tren cac anh xa tuy6n tinh Ta ki hieu tap cac anh xa tuy6n tinh tit khong gian vac to V dAn khong gian vac to W la Hom(V, W) hay Hom K(V,W) de chi rd K la tniang co sir. HomK(V,W) la met khong gian vec to tit truong K vbi hai phep town nhtt sau: Vai f, g e HomK(V,W); anh xa f + g e HomK(V,W) the dinh ben (f + g) (a) = f(a) + g(a) vdi moi a e V. Vol k e K, f e Hom K(V,W), thi kf: V —> W xac dinh boi (kf)(a) = kf(a) vdi moi a e V. Anh xa f + g dupe goi la tong Kin hai anh xa f va g. Anh xa k. f ddoc goi la tich eda anh xa f vdi vo hdong k. 3 - Di6u ki6n xac dinh anh xa tuy6n tinh Anh xa tuydn tinh f: V —> W hohn toan chidc :Mc dinh khi Mei anh cOm met co so. NMI le„ , e„) la co se, cOta khong gian vac to V va a,, a„ la n vec to cent MI:Ong gian vec to W (V, W la cac khong gian vec to tren clang rant trnong K), thi Kin tai duy nhdt met anh xa f e flomK(V, W) de f(ei) = a, yea j = I, f la don cdu khi va chi khi he , a„} doe lap tuyetn tinh, f lA clang cdu khi va chi khi he M I , , a„) la co sa oda. W. Gia = EJ la cd in ciaa W, thi f(e,) = E, = vaaha tran A = (ad goi la ma 62
  59. tan dm anh xa tuy6n tinh f dOi vOi co sa {e,) va {EJ. Nhg vay 16u cho cd sa E cua W va co sa e cua V, thi dinh 19 tren chring to :Ang co mOt song anh girla tap Hom K(V, W) va tap 1),E#t(m, n, K). FlOp thanh cua hai anh xa tuy6n tinh la mot anh xa tuy6n rhh, nghia la nOu f: V —> W va g: W —> Z la the anh 'ea tuygn thi g f: V > Z cling la mot anh xa tuygn tinh. 4 - Anh ca hat nhan cua anh xa tuygn tinh Cho f: V —> W la thing thu tuy6n tinh gifla cac khong gian 7ec to, neh X la khong gian vec to con cua V, thi f(X) = { f(a) I a E X) a khong gian con cua W, va n6u Y c W, Y la khong gian eon W thi f-'(Y) = e VI f(a) e Y} la mot kWh:1g gian con cua V. Ta goi Kerf = f-1 101 la hat nhan cda anh xa f va Imf = f(V) la inh cua anh xa f. S6 dim Imf doge goi la hang cua anh xa f, kf lieu rang E Gia sit f e HomK(V, W), la don eau khi va chi khi Kerf = {0}. Nth dimV Fa hfiu han thi IimV = dim Kerf + dim Imf. Cho ma Wan A e Mat(m. n, 1K), xem A nInt ma tran cua inh xa tuy6n Unh f: K n —> Km trong thc od sei chinh tdc. Khi do tang cua ma trail A (da dude dinh nghia trong chudng I) hang rang cua f va chinh la hang cua he vec td cot cua ma tran A. 5 - Ma tran aim t‘i thing cgu trong the cd ad khac nhau Cho f E HomK(V, W), trong co se: e = (e„ , e n) f cO ma Wan = (ad), nghia la f(e1) = Za ijei . i=1 63
  60. Gie sii & = (6o••ogs) le mot cd sa &bac, ma E, = /Cijej , tron€ i=1 cd s6 6, f co ma tren B = (b ii) ughia = Eb ii Ma trar i=1 C = (co) chide goi la ma tren chuyen co sa. Ta ca: B=C' AC Hai ma tran A ya B dude goi la deng deng neiu cú ma trey khong suy bign C de B = C -' A. C. Nhu yey hai ma Win cue ding mot phep bign (lei tuygn tinh trong hai cd sd khic nhau &dug clang. Ta goi vet min ma tren vueng A la t6ng cac phen to trer • &tang Oleo chinh. Hai ma tran tieing deng co vet bang nhau Vet cua mot to deng cdu tuygn tinh la vet cila ma trail cfla ni trong mot cd sd nal] do. Vet clad ma tran A dude ki hieu la trace A hay trA. V6t cua td d6ng cliu f thick ki hieu la tracef hay trf. Đ 3 HE PHUtiNd TRINH TUYEN TINH 1 - He pinking trinh He Ea kixi = bk (k = 1: 2, , i=1 dO aki , b k E K cho trUoc, k = 1, , m; it= 1, , n. xi la cac en, dude goi la h0 phudng trinh tuyen tinh (hay 14( phudng trinh dai s6 tuygn tinh) g6m m phudng trinh, n an so Khi K la truang s6 (nhu R hoac C), thi cac a to goi la the he 66, hQ se" to do. Ma tram A = (ak;) goi la ma tram cac he sg. 64
  61. a ll a12 aln b a b2 Ma trail Abs = a 21 a 22 2n goi la ma tran b6 sung, no cú ducic tii ma tran A bang each them cot cac he so' tti do vac, cat thu (n + 1). b Neu ki hieu X = va B = la ma tran cot, xni bini thi he phtiOng trinh (1) co the vieit duoi clung: AX =B. 2 - HQ Cramer He n phudng trinh tuyein tfnh n an s6, ma ma Iran cac he s6khong suy bi6n goi la he Cramer. HO Cramer c6 nghiem duy nhdt. each tam nghiem nhu sau: Cdch 1: Xet phasing trinh ma trdn AX = B, vi detA # 0 nen tan tai ma trdn nghich dao Al', va ta co: X = B Cdch 2: Xet he vec td cat an}, ma a i = (aii ,asi , ,a mj) va b(b„ , b0,) la vac to cat td do, the thi he viat dude dU6i clang =b. N6u ta goi Di la dinh thiic cda ma trail nhain dude bang each thay cat thit i cda ma trdn A bat cot cac he si6 tp do, thi x = ado D la Binh thiic cim ma tran A. I D 65
  62. 3 - He phtiong trinh tuygn tinh thuAn nhiIt HO phudng trinh tuy6n tinh thuan nhal ce clang: AX = 0 (2) Xet ma tran A = (ao) nhu ma Wm cua anh xa tuyen tinh f: K" —>. Km trong cAc co s6 chinh tac cua 1K'l va Km, thi tap hop nghiem cilia (2) chinh la Kerf. Mei cci sa caa Kerf goi la met hee, nghiem co ban cilia (2). HO (2) ludo cú nghiem x, = = x„ = 0, nghiem nay goi la nghiem tam thuong. KM rang A = n, thi he chi co nghiem tam thuang. Khi rang A = p < n, thi tap hop nghiem la kh8ng gian vec to n - p chieu (n la s6 an ciia phudng trinh). 4 - He phudng trinh tuye'n tinh tong gnat a) Dinh b./ (Gauss hay Kronecker - Cape 11i) He phudng trinh Za iixj = hi (i= 1, , m) (1) i=1 ce nghiem khi va chi khi rangA = rangA hs. b) Phining phap khet nem Gauss Cho he pinking trinh (1), n6u dung cac pilau Men dpi sau day thi La van nhan dude met he phudng trinh thong doing \TM. he (1), nghla la he cO ding tap hop nghiem nhit he (1). + :Than hai ye cUa met phudng trinh nao do cem hee, vat s6 k #0. + Geng vao met phudng trinh cua he sau khi da nhan met s6 bat ky vao hai ye cem phudng trinh khac.- + DAi the to cua phudng trinh cem he. 66
  63. Cap Oen bi6n ling during cl6i vOi he phuong trinh chinh la cac Olen bin d6i so cap thy(' hien Den cac clang dm ma trail 1)6 sung Abs cUa he. Dung phuong phap khil Gauss la Dille hien cac phen bign ckii Liking during de chia he phuong trinh (1) v6 he phuong trinh ma ma trail 06 dung: I 131 P 0 1 h ip p+i m-p 0 0 b'm p n-13 phan t>i a phan goch (*) co the khac 0. Khi rid n6u 13,;+1 + + > 0 thi he ye nghiem, n6u = b'„,= 0 thi he cú nghiem phn thuOc n-p tham s6. Đ4. CAU TRUC CUA T11 DoNG CAU 1. Khong gian rieng - da thitc dac thing Cho V la khong gian vec to tren truong K (1K bang K hoac C). Wit anh xa tuy6n tfnh to V d6n chinh no dude goi la mat to dOng calu tuy6n tinh. Tap ode tv ding cOu tuyeyn tinh eau V kY hieu la Hom K(V, V) hay EndK(V). Gia f e EndK(V), W la khOng gian con ena V; W chick goi la khOng gian con bkt bi6n elia V n6u f(W) c W. 67
  64. Vec to a # 0 thuec V dude goi la vec to rieng cim f e EndK(V) ling vat gia tri rieng X ngu f(a) = Ira, A e K. Khi do khong gian met chigu sinh bai vec to a Et met kh8ng gian vec con bgt bign cim f. Val A E 1K, tap ker(f - AId) khi no khac {(5} la khong gian con cim V, gam vec to khong va tat ca cac vec to rieng caa f ling vdi gia tri rieng X. Kitting gian nay goi la khong gian rieng cim f v6i gia tri rieng A, ki hieu Gia sit ta cl6ng eau f e End(V) trong met co sa Mao do cim V co ma tran A, thi det(A - XIn) IA met da fink bac n dovi v6i bign X, khong phu thuec vao vice, e chon co sa, va &lac goi la da thitc (lac trang caa ta (tang eau f (ta cling nOi do IA da that da, e trang caa ma tran A), ki hieu M.(x) = det I A - X In I . Nha vay A la met gia tri rieng cim f khi va chi khi X11 nghiem &la da that dac trang eim f. GM sit , Ak la cac gia tri rieng cigi met phan biet caa I Px Pik la cac khong gian rieng Wang ling via ale gia tri rieng do, thi t6ng Pr + Px2 Pxk la tong true tigp. 2 -Ong gian rieng suy Ong GM sit V la khong gian vec to tren (Wong K, f e End K(V). Vdi mei A e K, xot tap {a e VI co s6 nguyen m > 0 de1 (f-XIcl,)m(a) = 6). DO la met khong gian con caa V, khi khac {0} no &arc goi IA khong gian Hong suy rang cua f ling vdi A va ki hieu IA Ta thgy rang: 68
  65. + Vdi moi khong gian rieng suy rang 3 k 7r IA met gia tri rieng cim f va Y, c Rs. + V6i X IA gia tri rieng cua f, dim2h x bAng bei cern nghiem 7. cua da thitc dac trung /. f(X). + Mot g?",, la met khring gian con cila V bait bi6n qua anh xa f. + Gia sf1 , "f.k IA cac gia tri rieng phan biet tiing cap va u, e \ {0} = 1, 2, , k) thi he cac vec td {u„ uk} doc lap tuy6n tfnh. 3 - Tit citing clu luy linh a) Ta not ring f e EndK(V) IA tu ding caiu luy linh nau tan tai s6nguy'en k > 0 de'f = f o of= 0, hdn nua, nau # 0 va k hin 0' = 0 thi k goi la bac luy linh cua E Tu dang udu f e EndK(V) ma co cd sa te„ , en} sao cho Re) = (i = 1, , n-1) va Re p) = 0, thi hay linh bOg n va ed sa {ei, , en } dude goi 11 cd so xyclic d6i v6i f. Trong cd SO xyclic ma trail cua f co clang: 0 0 0' 1 0 0 1 0 1 0 0 1 0 69
  66. b) f e EndK(V). U la khong gian the to eon cem V, U chide gui IA khong gian vec to con xyclic chid vdi f nEu U IA f- bas biEn va trong co mot ca so xyclic dee vdi f/U: U -> U. c) NEu f e EndK(V), dimV = n, thi V phAn tich dude thanh tang trip tiEp cua cac khong gian vec to con xyclic doi vdi f. Vdi mdi so nguyen s 2 1, sE cac khong gian vac to con s chiEu xyclic doi vdi f trong moi each phan tich dEu bdng nhau va bAng: ran g(f 8') - 2rang(f s) rang(f d) N'Eu 9 ) IA khong gian rung suy rang cua f dng vdi A, thi (f - A.Ick) la td d6ng eau Iuy linh cern V. 4 - Ma tran clang chutha lac Jordan caa tat d6ng eau GiA sei V IA khong gian vec td hisiu han chigu tren truang K, f e Endx(V) ma da that ddc trung 94(X) cd dang: i(X) = (>1 (X , -X)`2 -X)sk (cac &Ai mat phAn biet, i = 1, 2, , k), khi do V la King true tiEp tha cac khong gian riAng suy rang V = x G3.? X2 e Xk va trong V co mot co sa (ei) = 1, n; (n = dimV) dA trong co ad do ma trdn din f tao bdi the khung Jordan 0 70
  67. nam doe (Wang cheo chfnh so khung jordan cAp s vai phan eh& X, bang: rang(f - - 2rang(f - kildv)s + rang(f - Ma tram clang tren cua f xac dinh duy nhAt sai khae each sap xap cac khung Jordan. Ma tran do dude goi la ma trail dang chuan tac Jordan cila tp (tang eau f. B- VI DV Vi d4 2.1: Xet R", R!" la the khOng gian vec td tren R. Cho x„ x2, , x„, la m vec td thuee R". ]E la met khong gian vac td con eila ChUng minh rang tap IF the vec td ena K" dang Ztixi (t1, , e E Fa khong gian vec to con ena K', dimF = dimE - dim(E nN), trong de N la khbng gian cac nghiem tha phtlong trinh: E ti., =0 (a do t i, tn, la cac An). i=1 Lai gidi : Xet anh xa tuyan tinh u c Hom(Rm, R") a do u(t,, t„,) = EtiXj . Khi do F = u(E), vi vay F la khong gian vec to con cna R'. dimF = dimE - dim(Keru'), u' = uI E. Kern' = (KerU) fl E = E (1 N. Nhtt vat dimF = dimE - dim(E n N). 71
  68. Vi du 2.2. Cho x i , , xi, la nhUng vec to khac khong trong khong gian tuy6n tinh V. Gia sit c6 Oleg bi6n d6i f e End V sao cho f(x1) = x1, f(xk) = xk + vdi moi k = 2, , n. Chung L6 rang he {x1, , xj la doe lap tuy6n tinh. Lb gidi: Ta chiing minh quy nap theo n. Via n = 1, thi {x1} dee lap tuy6n tinh do gia thiat x 1 # 0. Gia sit menh de dung vdi moi he fx„ xki; k < n - 1. Ta chimg minh menh d6 dung vdi k = n. Xet t6 hqp tuyen tinh ECiX i = 0 . (1) i=1 Khi do: ) = CIX ] Zcif(xi) = c,x, + Eci (xi +xi_1 ) = i=1 i=2 1=2 = ZeiXi . (2) i=1 i=2 n-1 Tit (1) va (2) to suy ra: Ecix,„ = = 0. i=2 i=1 Do gin thik quy nap hee, fx,, , doe lap tuyen tinh nen C2 =e2 = = cn = 0. Tit do suy ra c,x, = 0 ci = = = c„ = 0 Nhu vSy he {x1, , xj doe lap tuy6n tinh. 72
  69. Vi du 2.3. Trong khong gian vec to V cho he ak} cac vec to doe lap tuygn tinh ma m8i vec to dm no la t6 hop tuyen tinh cua cac vec to cim he 113„ , Hay ehung minh k 1. Ta se cluing minh ak phty thuec tuygn tinh. Xet t6 hop euyen tinh k 1 /Xiai =0 1=1 i=1 j=1 I ( k pi =o j=1 .i=1 EauXi =0 (2) j=1 vdi j = 1 do he pi leriic lap tuyen tinh. Vi he phuong trinh (2) la he phuong trinh thuan nhat, cú s$ an nhigu hon sg phuong trinh, nen no co ve s6 nghiem, nhu vay 73
  70. ton tai nghigm (x„ , xk) khac khong. Tn d6 suy ra he la,, ,a k phu thuOc tuydn tinh. Dieu nay trai vai gia thi6t. Do do k i E/E2 IA khong gian thacing va h: E, —> E/E2 li han chd caa anh xa chiou chInh tde troll E,. 1) Tim didu kien can va chi dd a) h 1a toan anh b) h la don anh 2) Chung tO khong gian (E, + E,)/E2 (fang eau v6i kholu gian Ei/EinE2. L of Rich: 1)a) X& E —> E/E2 la anh xa chidu chinh cac, h = h toan anh a ME,/ =11(E). E, + Kern = E + Ker n = E E, + E2 = E (vi kerb = E2) b) Ta en Kerh = E, r1 Kern = E, n E2 Nhtt vey h don anh a Kerh = 0 a E, E, = {O} 2) GiA. se F = E, + E2; Xet F F/E2 va k = 1-1/E, • Do phAn 1) k la toan anh va kerk = E l C Kern = E l 11 E2. Do do to co E l/El fl E2 ding can not E 1 2>E2/E2. 74
  71. Vi du 2.5. GM. sit V la khong gian vec td tren truing K. Ty Bong can p: V -+ V ducic goi la met phep chien ngu p 2 = p. 1) Chung to rang ngu p, q la hai phep chigu, thi p + q la phep chigu khi va chi khi pq + qp = 0. 2) Chiang t6 rang p.q IA phep chi gu khi va chi khi [p,qI = qp - pq la anh xa tuyern tinh chuye'n Imq van Kerp. 3) Vol p,q Fa hai phep chigu sao cho p+q la phep chigu, hay cluing to Im (p+q) = Imp + Imq va Ker(p+q) = Kerp n Kerq. Lei gied: 1) p+q la phep chigu ra (p+q) 2 = p+q p2 + p.q + q.p + = p+q p.q + q.p = 0 2) p.q la phep chigu (p.q)2 = p.q.p.q = p.q = p2 q2 q (pq — pq) (Imq) = 0 Er> p (qp — pq) . q .(=> (qp — pq) (Imq) c Kerp. 3) Vi mai phep chigu p co rang p = trace p, va vet cim tong hai anh xa tuygn tinh bang tong cac vet cim no, cho nen ngu p + q la Agri chigu thi: trace (p + q) = trace p + trace q. va rang (p + q) = rang p + rang q. Tit do dim Im (p + q) = dim Imp + dim Imq, suy ra Im (p + q) = Imp ED Imq. D6 °hang minh Ker (p+q) = Kerp n Kerq to nhan thay kerp nKerq c Ker(p+q). 75
  72. Nguuc lai vdi x e Ker (p + q) thi p(x) + q(x) = 0. Do Im (p.+ q) = Imp $ Imq nen MI p(x) + q(x) = 0 suy ra p(x) = q(x) = 0, hay x E Kerp n Kerq. Vi du as. Gis sit E va Fla hai khong gian vec td tren trulang K; Horn (E, F) IA tap cac anh xa tuyeal tinh tit F Mn ]F; Ft la khong gian vec to con cna F. a) Chiang to rang Ê = e Horn (E, F) / Imf c F 1) la khong gian con cria Hom (E, F). b) Gia. s5 F= K la mot phAn tich cua F thanh t"o"ng tryc tip i=1 ciaa khong gian F. Vbi mdi F, xet 2, = if E Horn (E, F) I Imf c Fd. Chung to rang Horn (E, F) = 2. WL gia'i: a) Do g ding kin vai phep town tong anh xa va nhan anh xA vdi mat vet hriong thuac K, nen hieln nhien la khdng gian vec to con crIm Horn (E, F). b) Vdi h E Horn (E, F), vbi m6i x e E, to phan tich h(x) = y theo eec thinh ph'An y = h(x) = Eyi , yi e F1 i d Xet E -, K, e Horn (E, Fi) = 2 hi(x) = yi 76
  73. Va vdi moi x e E, thi h(x) = Eh; (x) i=1 Do 05 h= , e Gia six Eh; = 0 nghia la vial moi x e E, ( ) (x) = hi (x) = 0, h,(x) e F, 1=1 1=1 to do suy ra 11,(x) = 0 voi moi i vi F = e K. I=1 Nhu \ray Horn (E, F) = e g. =1 Vi du a 7. GO Q la Huang s6 h> u ti va WI la mot tkp hap khong rkng. Ki hiku E la tap cac anh xa tif c32 vao Q. E la Q - khong gian vac to viii hai phep than: f, g e E, a e cc9, k E Q the (f + g) (a) = f(a) + g (a) K. f) (a) = k. f (a). Gia sif V la khong gian vac to con ciaa E. 1) Chang to rang: nku f e V va co n dim x„ x„E Ca sao cho ingui=j . . — k(x,) = = ej=1,n Ongui#j thi he 10, •••, f,,} dOe lkp tnyen tinh. 77
  74. Goi W la khong gian con cea V sinh bed f„), khi do m81 g E V 11611 viet dude met each duy nhat dudi clang: g = h, +112, a do h, e W, h2 E V va. h = 2(x)= 0 yea mai i = 1,n. 2) Chung CO rang hai tinh chat eau day la Wong clueing: a) dim V n b) Ten tai n ham g„ thuec V va n diem x„ x„ ciaa W( sao cho g, (x,) = 8,; vet moi i , j = I,n . Lai gidi: 1) Xet to hop tuyen tfnh EXif, = 0, c Q. 1-1 Khi do vdi moi x E cam , LW; (x) = 0; Chon x= x„ thi (xi) = 7.j = 0 ( = 1,2, n) suy ra he {f„ , f„) doe lap tuyen tinh. Val g e V, ham h, e W cAn tim phai thaa man g(x) = (h, + h2) (xi) = h r (x,) vdi moi Do If„ f„) la co sa cem W nen h, :11 1 (x, = EgfrA . lei 1=1 Dat ha =g - h, eV thi h2(xj) = g(xj)-112(xj) = 0 vdi moi j = 1 n. 78
  75. 2) Theo phan 1) neu b) clime th6a man, thi he (g„ g„) doe lap tuyen tinh, do \ay dim V 2 n; nglila la menh de b) a) thing. Ta chung minh a) suy ra b) bang phydng phap quy nap theo n. Vdi n = 1; n6u dim V 2 1, tan tai fi x 0 va vi vAy cú x, e c14 de fi(x,) = A x 0. Chon g, = thi (x,) = 1. Gia sii menh de dung vdi moi k = 1,2, n. Ta chiing minh nd dung vdi n 1. Theo gia thi6t quy nap dim V n + 1 dim V ?. n nen tan tai n ham g, E V va n diem e A de gi(x,) = 8,, (i, j = 1,2, , n). Goi W la khong gian con sinh bid {g„ gn}, to cd. dim W< dim V. Theo cau 1), vdi f e V \ W, to cd f = h, + h 2, vi h, e W nen h2 0; vi th6 c6 x,,, d'e' h1 0; (x,„., x x, vdi i = 1, 2, , n). h2 - hi bin+, vdi MOt i = 1,2, , ) t n+1. Bay gia = g - = 1, , n), thi vdi j = 1, n to co gi (xj )= (xJ )= si; va gi (xn+1 ) =g. (x„. 1) - x; = 0 n6u 2,1 = gi (x„,)_ Nhu Nty co n4-1 ham {g-i } th6a man digu kien bai town. Vi du 2.8. Gia sit 98 la ho dem doe cac anh xa tuy6n tinh tit R" (16n R"I; vdi mai a e R" xet 0(a) = {f(a) / f e :43}. GM sit g la Anh xa tuyein tinh to den õR" sao cho g(a) E :0304 vdi moi a e K. Chung minh rang g c PC. 79
  76. Lo gidi: Ccieh 1. \TM m6i x e R, xet vec to dx (1, x, x2, H9 eac vee to dx c6 tinh chat: a) vdi x„ doi mat phan biet, thi ta xi dPe ld tuy6n tinh, vi Binh # 0 (Dinh thite Vandermonde) b) V6i m6i dx e R', tan tai f E sao cho g(Ci;( ) = f( ). Ta phai chiing minh g c 93, nghia la c6 f e d f g@t„i ) tren bb la co so cua R". Gia sii node Lai, vdi m6i f E c6 khong qui (n - 1) sd the phan biet xi cl6 g(d xi ) = *xi ). Do hop &dm Mtge eac tap bop hfiu han phan to 1a mat to hap kh8ng qua deM dude, nen sod cle vee to d x e ma g(d x e g3(ax ) le killing qua d6m dude. Digo nay trai vdi gia thie Nhu vay phai co co sa Vt xi , ,a x ) oda an vti 06 f E tgi d = f(dixi ) vOi moi i= 1,n; nhu vay g=f e 80
  77. (rich 2. (Mtn}, cho doe der hied mot sa sri kien cilia khong gnin metric). Theo gia thiet vdi moi a e der f e de" f(a) = g(a). nhu vay — g) rad= 0 => a e Ker (rig). NInt vay U (Ker (f — g) f E gq} = R. Hia sir g vSy vdi mai f E thi f — g x O. do vay dim Key (1— g) n — i. Vi vay Ker (1 - g) lie met kliong gian con thing, khbng dau tra mat cent R". Dieu nay gay nen man thuan do RS la khong gian metric day vdi metric dieing Hwang vii dinh 19 Haire aid rang: met khong gian metric day kheng the bang hop dem dude mita nheing rap hop khong dim Hu mat. III. du 2.9. Cho hai so nguyen dining r, n ma r < n: (it c Mat (Th R). rg." = rank rang cl = r m& (CP = Hay chdng minh vet ()f = a,, + a i„ + + a„,, = r. (Vet can ma Man red thudng duo( ki Heti bdi trace cel). Xet f e End (IR") co ma trhn trong cd ed tly &nen e = le,, e„). i(e,)= apie Theo gia Hired to cd f2 = f. Bat Y = - f(a) I a e khi de Y c Kerf, nen Yea a e thi x = a - 1(a) e Keil, do vay a = f(a) + x e IMF+ Karl 8 1
  78. Ta co Imf n Kerf = {0), that Gay, gia sit p e Imf Kerf, thi co a e R" d f(a) = p va do 13 E Kerf nen f(0)= 0 suy ra: f(p) = 12(a) = f(a) = p = 0, vay Imf n Kerf = {0). Nhu vay R" Imf O Kern dim Imf = r va f I Imf = id, Chon ed sa s Rua R" sac) cho {E,, Ê,.} c Imf, {E„,„ E„) e Kerf, thi trace f= trace al= r. Cho Nhaat lai rang n6u f la anh toa tuyeat tinh cie'n Jrco ma Ran oat = (ad trong cd sa e = (e,, nao do ciaa Tthi so' a„ +, ,+ a„„ khong phu thuoc vita vice chop cd so oda °L va clack goi la vat ciaa anh xa tuyan tinh f va chiac kf hieu la trace f. Vi du 2d0. Gia sa °W la hai khbng gian vac td treat twang K, f c HomK CP; °Tf). Xet anh xa f: Kerr Gil" [a]-a f [al = f (a), Hay chttng to f la dun eau tuyen tinh va Imf = Imf , ta do f la dang cau to /Kerf len Imf. ianh xa f lh sac dinh, khong phu thuR vao dai dien. That way, vdi [al = thi a - a' e Kerf, W do f(a) = f(a") hay II-al = f thd thay f la Maya tinh va Imf =Imf.Ta chung to f la ddn cau. That Ray RR [a] # [a'] thi a - a Kerf f(a - = f(a) - f(a) # 0 do do f(a) # f(a) suy ra f [a] # f Nhu way f don cau va do do f la clang cau tit cliKerf len Imf. 82
  79. Chu" y: neat so chieu Irau hen, thi tfx vi du Hen to dim tillierf = dim Imf say ra dim "P= dim Kerf + dim Imf. du 2.11. Giei he phudng trtnh +2x 2 +3x 3 +4x 4 =30 -x i +2x., - 3x 3 + 4x 4 -10 x, - X3 +x 4 = 3 x i + +x3 + X 4 =10 gthi: 1 2 3 4 -1 2 - 3 4 D = = -4 0 1 -1 1 1 1 1 1 Day 1a he phudng Huh Cramer. 80 2 3 4 1 30 3 4 10 2 - 3 4 -1 10 -3 4 1), = D,= = -8 3 1 -1 1 0 3 -1 1 10 1 1 I 1 10 1 1 1 2 30 4 1 2 3 30 - -1 2 10 4 -1 2 -3 10 Da = -12 D'= = -16 0 1 3 1 0 1 - 1 3 1 1 10 1 1 1 1 10 NMI vey , = 1, x 2 = 2, x3 = 3, x 4 = 4. 83
  80. Vi dii 2.12. Gihi va bran luhn theo thaw s6 A.X1 x, + x• 1 Xx, + x. 2, + 4x 3 I) = a) Veil X s 1 vic a x -2. Day la he Cramer. it +1 1 va ta ce, - , x - X +2 2, + 2 +2 b) vdi A = 1, to c6 he Wring during vdi: x i + x2 + x,, =1 hay x3 = I - x, -x., do x i , x, lay trtyl, c) Vol X = -2. He co dang: -2x 1 + x., + x. = 1 xi 2x, + x3 g - 2 xi + x, - 2x 3 1 GUng vg still ve curt ba planing trinh Ln c6 0 = 3, nhn vhy he voi nghiam. 84
  81. Vi du 2.13. Dung phitong phap khn. hay giai he phuang trinh: xt + 3x, + x 3 + 2x 1 )(5 = 2 3x 1 + 10x 9 + 5x 3 + 7)( 4 + 5)(5 = 6 2x 1 + 8x, + 6x 3 + 8x 1 + 10x5 = 6 (I) 2x 1 + 9x„ + Sx 3 + 8)41 + 10)( 5 = 2 2x 1 + 8x, + 6x 3 + 9x + 12x 5 = 1 Lm giai: Nhan hai ye aim phtiong trinh (tau vdi Inning so/ thich hop, I.()) acing vac) eac phtaing trinh khae, ta clia; he pinning trinh ()rang throng vdi he (I): x + :3x 2 + x,, + 2x + X5 - 2 X9 + 2X3 x 4 + 2x5 = 0 2x, + 4x 3 + 4x 4 + 8x5 = -2 (II) 3x, + 6x3 + 4)( 1 + 8x 5 = -1 2x, + 4x 3 + 5x 1 + 10x5 = -3 Nhan pIntong trinh thit hai cna he (II) vdi gag s6 thich hop roi tong viva du; phuong trinh kink cita he, ta &toe he Wong dyeing- x 1 + 3x 2 + x 3 + 2x 1 + x5 = 2 (I) x, + 2x 1 + x, + 2x5 = 0 (2) 2x 4 + 4x5 =-2 (3) (Iii)( x 1 + 2x5 = -1 (4) 3x 1 + 6x 4 -3 (5) 85
  82. Car phtfting trinh thit (3), (4). (5) trong he (III) la tudng during. Vi 04 he (III) tudng during vol. he x i + 3x2 + 2x 4 2 y 2 + 2x 3 + -x I + 2x 5 = 0 (IV) x 4 + 2x 3 = -1 Giai he (IV), to (little xn = —1 — 2x 5 X2 = 1 — 2x. a do x4. x Inv 31 xi = 1 + 5x i + 3x 5 Vi dv 2.14. Cho hai ma lien A, B thuOr Mat (n, K), A = (ad, B = 09. Khi do A + B =(a i +1)0 (bloc goi la tang hai ma tren A va B. Chang minh rang: 1) I rang A -rang B l < rang (A + B) a rang A + rang B. 2) rang A + rang B -n a rang (A B) a min (rang A, rang B) 3) Nan A' = E, tin rang (E + A) + rang (E - A) = n. (3 do E la ma tran don vi cap n). 1) Gia stir f, g la hai ph&n tit cilia End (õ "), co ma 'Iran A, B Wong fing trong m(llt cc; se( s = (s 1) di( cho. Khi do f + g c6 ma tren A + B. VI Im(f + g) c Imf Img, nen: .dim(lm(f+g)) < dim(imf + Img) dimlmg. 86
  83. Tit de suy ra: rang(A + B) < rang A + rangB. Mat khdc: rangA = rang(A+B-B) < rang(A+B) + rang(-B) suy ra: rangA < rang(A + B) + rangB Tit do: ningA - rangB < rang (A + ta: rangB - rangA < rang(A + B). VI yay rangA - rangB 15 rang (A + B) 2) Ta co f: K" , g: K" la hai anh xa tuy6n tinh, Im(f o g) = Im(f img) c Imf nen rang(AB) < rangA. Mat khan: rang(AB) = dim(lm fog) < dimlm f = rangB. Do yay rang(A o B) < min(rangA, rangB). Bay gio ta churig minh rangA rangB - n < rang(AB). Ta co dim lle = n = dimIm(f o g) + dimKer(f o g). Mat khde 26t anh xa Kor(f g)/Kerg Kerf (1 Img a do tI/4xl = g(2), yin X E Ker(f 0 g). De they (To la (Tang au tuyeal tinh. Vi dimKer(f o g) = dimKerg + dim(Kerf fl Img). Tit do dim Im (f 0 g) = n - dim Ker(f o g) = dim Img - dim(Kerf Img). 87
  84. Nhu vay: rang(AB) = rangB - dim(Kerf (I Iing) je rangB - dimKerf rang(AB) -e rangB+rangA- n. 3) Vi A2 = E (E la ma (ran den vi), nen: (A - E) (A + = 0 Vt vay, then phan 2), to co: rangA -E) + rang(A + E) - n <0 hay rang(A - E) t- rang(A + E) n. M3t khac, then phfin 1) rang(A+E) + rang(A-E) 2 rang(A = rangA = n Do vay rang(A + + rang(A - E) = n. V( du 2.15 Cho ma tran A e Mat(n, R), cac phan to (ran duang cheo chinh bang 0, con cac phan to khac hang 1 hoac bang p, d (16 p la mot eel nguyen lan hdn 2. Chung to rang A e n-1. 142i )(et ma tran II = (h ij ) e Mat(n, 1R), .3 do b ij = -1 vol moi i, j. Khi do rangB =1. va A + B = (CO, vdi C,, = -1, con thc phfin to khOng thuOc during cheo chinh bang 0 hofic bang p - 1. Nhu vay det(A+B) = (-1)" (mod(p-1)). 88
  85. Do p-1 > 1 nen det(A +B) # 0 hay rang(A+13)= n. NhUng Chao vi du 14, to co: rang(A+B) V la Di dOng cau clic( C - khon g gian vec (5 V, 11611 ce mot vec td Hong e t Bat V, = Vect , thlkhong gian v@c t d con 1 chi6u (p - brilt biOn. Gia sv (ili xily dung duo() car khong gian con (4) - bat Heal V,. V2, Vk ma V, = Vect , V2 = VITA. Vk = Vect e h 8: dimVk =k. Tel get long cau Wk : yv k \wk. [U. I H> [(P( )] 89
  86. cam sinh bai clang cau p. Do tir k IA ta clang can caa khong gian vac to V/Vk tren truang se" phtic C, nen Co vac td rieng rea +i ling vdi gil tri rieng can chfing minh: Ubl Vect le, ,e k , e k*, I khong gian con k+1 chieu dm V va la - bat Man. kik Xet re hop tuyan tinh ei = 0 . i=1 Neu ?Lk+, 0 thi ek,i e Vk [ ek+1 = [6], trai vei vice k chon{ a k,, I la vac td rieng, vay 2 k,l = 0 tit do =0 nen iat = 12 0 do he {a; 6 k doe lap tuyan tinh. Nha way he lei , eki doe lap tuyan tinh = k + 1. Ta chUng minh Vkk i lap - bat Bien. Chi can chang minh: {14 ) e Vik*I- Ta cif) im(rp )1 ) 1 pk k- k+lk = ` T‘ -k+1 kk = •`k-hlk k-kk = `- k+1 k p (e +, ) - X e k+, e Vk C Vk+l, tit do suy ra 6 k+1 ) e Vk-kik Bang each do, ta say dung clam cat khbng gian: Vo = 0 c V, c c Vn = V p - bat hien Nhain xet: Ma tran tha ta clang cgu p trong co sa 1 , e „} )(ay clang a tran c6 clang tam giac tren. Dung ngan ngil ma [ran, ta c6 the phat hie: moi ma tran phew vuong cap n dgu d6ng Bang vOl mat ma tran tam gile tren. 90
  87. Vi du 2.17. (Dinh It Hamilton - Cayley) Cho V la mot K - khong gian vac to; cp e End(V); 9 c6 ma tran A trong cd so e = (ed i = 1, n; 1,, - la ma trail dun vi, khi do det(A - xI„) IA ma da thdc bac n, doe gut la da thitc da, c trung cUa tii (ItIng cau cp. D6- thay da thuc nay khong phu thuc van ed so e = (u) da chon. Ta cling goi da tilde tron la da HI& dac trung dm ma trail A. Gia sit det(A - xI„) = a ox" + alx"-'+ +a„_,x +a„. Khi do to co: a0A" + a,A"-' + +an_,A +a„ = 0 hay 42'1 + a,""-1 + + an.Id = 0. Lai gidi: Xef ma luau B la ma trail chuyi1n vi ci'm ma tran phil hop am A - xI„. Cac phan ;I'M ma Iran B IA nhfing da thilc cua x v6i he tiI trong truong K, c6 hac khong qua n-1. Ta B = B0 +13,x + .„ + x"-2 + Bn_1 . x"-1 trong do Bo, Bi, B,1 la nhung ma trait vuong cap n, khong phu thuiic x. Theo tinh chat dm ma tran phu 110p, to co: (B0 + 11,x + +B R,. x"1 ') . (A - xI„) = det(A - x1,). = = (aox" + + + an_ix + a„). I„. Ta co mot he cac clang thitc sau: B0 A = an. I. - Bo = and • In • B2A - B, = an-2 • In. 9 1
  88. B„-,A - 13,12, = a, 1„. -B„., = a,,T„. NhAn vg voi vacua the clang their trcn Bin Met vdi I n, A. X- A" rOi enng Jai, to (kith an A n + + a, ^ A+a n I,= 0. Chu Dinh 19 Hamilton-Cayloy thadng dude ghat binu: MO ma tran vuting dela la nghiem maa da tithe dar trnn, ua ne. Vi rig 2.18. GiA s t B la ma trap ley linh, la ma tran giro halm vdi B. Hay chtIng mink det(A + B) detA. Leh gidi: a) Xet twang hen detA = 0. cia stY n la bac lily linh rem B, nghia la Bn = 0. Ta re: (A + B)" = zc nk A k B n-k = Ecnk A k w _k = A Ick A k-1 n k k=0 k=1 k=1 Vi detA = 0 nen det(A+B)n = 0. Vi way det(A + B) = 0 = detA. b) Ttheng Mk) detA = 0. Do AB = BA non BA-' = A-1 B. 92
  89. Theo gii thiat B lay linh, nghla Isl co se n de 13" = O. tit do -1 B)" = 0 vi vay A-1 B lay linh. NM( vay tan tai ma tran kheng ay Men C do C - 911-1 B).0 la ma tran dime tan hal nhning ma An inking Bang: ' 0 o 1 0 1 0 1 0 am riot thco during cher) chinh. tr.( do ta co det(1 + A -1 B) = 1, a do la ma Ban don vi (tang dip vdi B va A. Nhu vay det(A+ B) = det[A(I + A-1 13)I= detA. (-Iot(I +2.111 B)= detA. Vi dy 2.19. GiA six V la met khong gian vec td tren truang va f e End,(V). Chiang minh rang Kerf = Imf khi va chi khi = 0 vh ten tai h e Enci,(V) do h o f+ foh= Hy. Didu ki(es chi: Tu f = 0 say ra Imf c Kerr. Bay gia ta chUng minh Kerf c Imf. VOi IV la Itheng gian ni bat 14 cna V, to co: (1) 0 f + f 0 II) (NV) c hof (W) + f 0 h(W). Do ta co of + f oh= Id, nen W c ho 1(W) + foCuh(W) vdi mei W c V. Lay W = Kerf, thi f(W) = 0 WI ta Kerf c f o h(Kerf) c Imf. Nhu vay Karl= Imf. 93
  90. Digit ki•en din Gia si7 Imf = Kerf = V,. Goi V2 la phein bet tuygn tinh ego V, trong V nghia la V = V, $ V2; pl : V VI la phep chi& chinh tac, nghia la ven x e V, x = x, + x 1, x, e Vi thi p,(x) = xl . Xot anh xa ?: V2 -) x2 1—› 1 (x2) = nx2). De thgy If la Bang ego. Dat h = I -1 o P I , to en: f 0 h(x)= f -'. pi(x) = pi(x) = xi. Thong tki h f(x) = ho f(x, x2) = ho f(x2) = -I 0 Pi 0 f(g2)= f((2)= -1 • ?(x) = x.2- Do vgy (10 h + h (x) = xl + x2 = x. Do do foh+hoN Id o. Tu gia thing Imf = Kerf suy ra f 2 = O. Vi du 2.2a GiA si V la khong gian vac to tit' truong 1K. v? f e EndK(V). Vol m61 da thfic P e IK[x], P = ±a ixi , dat P(f) =0 'a; f' thu6c Endk(V). De thyIKPc] EndK(V) (13(P) = P(f =o la mat d6ng cau tuyen tinh. Da thitc Pf &toe goi la da thew t6i tie'u cum to d6ng au f new P1(0 = 0 vix n6u Q(f) = 0 thi Pf la vac cua da thew Q, voi Qe IK[x]. 94
  91. 1) Gia sia V I la kh8ng gian con oda V bar bin do=i vdi f va f, = f I V, la clYng au cam sinh. Chung to rang da thfic to) tiaai P, ena f, la Pdc cua da thfic to) tiPu P r. 2) Gia sfi rang V phfin tfch throe thanh tong cac kheng gian con baat bign del vdi f: V = la dying (Au cam sinh trim =1 V, va P, la da thuc tot tiPu cPa f,. Chung LO rang P, la boi chung nhO nhavt cua cic da thfic P. Liii gidi: 1)Vi f = f1 nen vdi moi Q E K[X], La co Q(f) = WO fit VI. kreh Pf lA da thfic ten tiUu rim f, thi Pr(f1) = 0. Do vhy Pf la bOi da thfic toi ti6u P, oda f,. Ta cling nhyn flay rang P, khang nhal thi6t la uoc thpc sp Lim P,, khi V, la klieg-1g gian con thpc so cim V. Vi du ne).1 '= %H", thi La c6 Pf = P, = x - vdi moi V, c V. 2) V, la khong gian con f bi6n, nen do can 1), da thfic :61 thi6u Pr chia hect cho vdi moi i = 1, 2, p. Do vyy Pf chia Mich° bOi chung nhO nhiit P cua Pi. Ta chUng minh P(f) = 0. That vyy, vdi moi i = 1. p to c6 I) = Q . P, tit c16 vdi moi x c V, x = Ix ; , I'(f)(x) = LP(f)(x,) = 1 =1 1=1 P p P(f; )(x i ) = EQ; (fi )13; (fi )(x ; ) = 0 do Pr(f) = 0 vdi i. r=-1 i=1 NMI vyy P(f) = 0, to do P chia 116t cho da thik toi tiPu Pf ty dang can f. 95