Tài liệu Bài tập địa số tuyến tính

PHAN HUY PHIJ • NGUYEN DOAN TUAN BAI TAP DAI SO TUYEN TINH NHA XUAT BAN HAI HOC QUOC GIA HA NOI Chin trach nhiem xual bcin doe: Gicim NGUYEN VAN THOA Tong bien Op: Bien tap: NGUYEN THIEF N GIAP 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. 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 có 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 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).