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PHAN HUY PHIJ • NGUYEN DOAN TUAN BAI TAP DAI SO TUYEN TINH NHA XUAT BAN HAI HOC QUOC GIA HA NOI rvikic LUC Chubhg .1: DINH THOC - MA TRA:N 7 A - Tom tat ly thuyeet 7 §1. Phep th6 7 § 2. Dinh thitc § 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 §1. Kh8ng gian vec to 57 §2. Anh xa tuyeen tinh 61 § 3. He phydng trinh tuy6n tinh 64 §4. Can true caa tai ding cku 67 B Vi dtt 71 C - Biti tap 96 §1. 'thong gian vec to va anh xa tuyeen tinh 96 §2. He pinking trinh tuy6n tinh 104 §3. Cau tit cna melt tu thing calu 106 D. Illidng sign ho(tc clap s6 110 5 §1. Khong gian vec td va anh xn tuyin tinh 11( § 2. He phudng trinh tuyeit tinh 12'; §3. 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 §1. Dang song tuy6n tinh aol xUng va dang town phuong 139 § 2. Killing gian vec to gent 135 §3. 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 Chuang 1 DINH THUG - MA TRAN A - TOM TAT Lt THUYET §1. PHEP THE Met song anh o tit tap 11, 2, met phep the bac n, ki hieu la '1 2 3 \ G I a2 G n} len chinh no duet goi la 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 the cem a n6u s6 - j) (a, a) am. Phep the a &foe goi la than ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6 - nghich the ciaa a le. Ki hieu sgna = 1 neM s la phep the chan -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„ • - • , i k doi mot khac nhau dr coo = 12 , coo = i3, a(ic) = i1 7 va a(i) = i vdi moi i x i„ i k . 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 ••• , ik) phan tfch chive thanh tfch 0 1 , xfch § 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 • phan tfch thanh tong, thi c) Ngu mot Bong (hay mot dinh thitc dU9c phan tfch thanh tong hai dinh thfic, cv th6: f an de = det al; a 21 21 a2„ a,,, + ani ‘ a n„ a ll an, ail a21 +alci ...a1,„ + de t all a21 —a 1111/ d) Cho A = (Ito) E ...a2 n " S ' Ill " S IM / Mat(n, K), thi = b) a do = aij &toe goi la ma tran chuy6n vi cim A. Ta co detA = detA t. 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 có 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 a ikAik /=1 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), va Oh ik), 1 s i, <1 2 < . < 1 =11 < j2 < ••• a co nghich the'. Vi vay, do so' nghich th6 caa a la k < , nen ten tai i o dg oc ii) < a i0+1 ; 0„) trong do 111 i = a ngu i # i„, i„ + 1, Xet hoan vi p = con p io = a ;0+1 , p i+, =a,„ thi HI rang 13 co nhigu hon a met , nghich the". Nghia la s6 nghich th6 ciga p la k + 1. 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 1 0 B= t1 vi C= 1 11 ma detB =1, detC = 1 nen detA = detB. detC = 1. Vi du 1.9. Hay tinh cosa 1 1 2cosa = 0 0 1 0 0 0 0 0 0 1 2cosa 0 2cosa 1 1 2cosa 21 Lai gidi: Khai trim dinh thac Lheo cot cub" to co D„ = 2cosa . - 17,1 _2 De thay D, = cosa. 1 = 2coi2a - 1 = cos2a. 2cosa, cosa Gia sa D1 = cosia \TM moi = 1, . k. Ta có 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„ = 1 0 + a-9 1 e" +e 1 1 0 a 0 N x0 1 1 eP + e -(1) 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 Khai trin theo cot tht nhEt, to c6: A n = (e P +e -P)A n _i: e 21' - e -2(P Nlinn xet rang 4 1 = 6 9 +Cc = A2 = - ((i 6 e e ro 36 - -39 e (P - e (1+1)6 - e -(lrv1),p Girt sit AR - e (-0 -e , (P k - 1, 2, , n - 1. Ta c6 An = (e c e -P)A n _ i -A n_, =(eP +e w)e nip - e npe( n-1) n4) ew Nhu v 6. }.: An = e - e (:+1)o - e -(n+1)4' — e q) - e (n+1)p - e -(n+1),p e 1 -e Vi du 1.11 Tinh: ll 1 D = dot 1 a1 a, an a l: +h i a, an a1 a, +b., an a1 a.9 ... a n +b n 23 Lai gidi: LAY ciOng dAu nhan vOi -1 r6i Ong vao 1ai to có ngay D = b, 1) 2 ...b„. cac thing con , Vi du 1.12 Cho da thric P(x)=x(x+1)...(x+n) Hay tinh Binh thdc: P(x) P(x) P(x +1) P(x +1) P(x + n) ... P(x +n) d= P(n-1) (x) P th-1) (x+1) P thl (x) Pthl (x +1) P th-1) (x + n) P (n) (x +n) gidi: Ta b6 sung de' dude ma Han dip (n+2): P(x) P(x +1) P(x +n) P(x) P4x+1) P4x+n) D= P (n ) (x) Pthd(x +1) P0P(x + n) pg+0( x +n ) P(„+l) (x) 1301+1) (x +1) 0 0 0 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 có clang: poc + 0 + k=1 24 P(k) (x +0.(x +11.) k k! ok = n). 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 P(x+n) Pkx+1.) d = del D - (x+n)°+i (n+1)! PTUx +1) . PT+I kx+1) P(n)(x+n) 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 + a l , , (n+1) tx„+ a do a l la hang s6 &do do. Khi nhan (long cue"' ding voi al (n+1)! r6i Ong vao clang trail no, ta dtta ma trgn ( ) va dang P(x 0 ) P(x] ) P (11-1) (x 0 ) Pth-e (x l ) (n+1)!x0 (n+1)!x 1 (n+1)! (n+1)! P (x i) ) PT-1) (x n ) (n+1)!x n (n+1)! (*) 25 Deng this ba tit dUdi len caa ma Dan (*) co dang (n+1)! 2 (n +1)! 2 x n +a i x n +a., xo +a i x o +a,,..„ 2 2 Cong vac) (long nay hai clang °Ma sau khi nhan voi cac s a, a1 ta nhan dude clang va (n+1)! (n +1)! (n+1)! 2 2 xo , (n+1)! (n +1) 6 9 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 = (n+vn x on Xn n-1 X0 x n1 (n+1)! k=1 IC ! X0 n(n+1) (-1) 2 .((n +1)!6'1 J k! k=1 26 .D n .