BÀI tập đại số TUYẾN TÍNH đh QUỐC GIA hà nội

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 A - Tom tat ly thuyeet §1 Phep th6 § Dinh thitc § Ma tram 10 B - Vi dn 12 C - Bei tap 35 D HtiOng dein hoac clap so 43 Chudng 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 § 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 §1 Khong gian vec td va anh xn tuyin tinh 11( § 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 § 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 Chuang 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 \ G I a2 G n} len chinh no duet goi la 15 del a, = a(1), 02 = a(2), , a„ = a(n) Tap cac phep the bac n yeti phep nhan anh xa lap met nhom, goi la nh6m del xeing bac n, ki hieu S S6 cac Olen t3 cua nhom S„ bang n! = 1, 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 = 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 dai k n6u c6 k s6 i„ • - • , i k doi mot khac dr coo = 12 , coo = i3, a(ic) = i1 va a(i) = i vdi moi i x i„ i k Vong )(felt dttoc ki hieu IA ik ) M9i phep th6 dau &tan tfch the tfch nhung yang xfch doe lap Met vOng xfch dal dude goi IA met chuygn trf Vong ••• , ik) phan tfch chive tfch , xfch § DINH THUG I Gia sit K IA met trueng (trong cuan sich 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) 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 Tinh eh& ceta Binh that a) Neu dgi cho hai dong (hoac hai cot) nao cim ma tram A, thi dinh auk cim no ddi da:u b) N6u them 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 • phan tfch tong, thi c) Ngu mot Bong (hay mot dinh thitc dU9c phan tfch 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 = aij &toe goi la ma tran chuy6n vi cim A Ta co detA = detA t 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 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), s i, 13 3.3 a) Q-602 -i612)2 + 16 673)2 + -+ n+1 2n 67 :32 That \ray, dat y, = x, + —(x2 + x„), thi -3-x,x ) ;2 Dat y2 = x2 + (x3 + + xn), tu f = +71 1374) + - Ex; +—xi x ; 3 i0 nhung (x, x) Ta thky x, y dec lap tuy6n tinh va vat z thuec khong ;tan vec to sinh bat x, y thi fez, z) = f(ax I by, ax + by) = a2 f(x, x) + f(y, y) + 2ab f(x,y) ?0, (") Theo gia thiet n6u f(x, a) =0 thi x cling phitong n6u f(y,a) = hi y cane phtiong voi a Nhung x, y khong tang phuong, nen (x, a) va f(y, khong deng that bang khong (**) Do co hai thvc k, I cK cho k' +1' > va k f(x, + I f(y,a) = TU f(kx + ly, = Theo gia thiet kx + ly ding phucing 'di a, trudng hop f(kx + ly, kx + ly) < Coat nhan 'cot (*) Do to a= 'K i x + ltyx Theo gia thiet f(a, a) = k, 2f(x, x)-11,1 f(y, y) = O )o f(y, y) > 0, f(x, x) '2 nen to co 1, = Ira f(x, x) = Nhtt t = k, x va ter [(a, y) = 0, f(a, x) = O Mau thuan 'got (*") 3.14 Xet A e M:11 (n, c Mat (n, C) Vi A phan di51 ximg len ma Iran iA Hemnit - Min vt ao) Ta co det (A - kin) = let (iA - ixIn) = Nhung mot nghiem da' c trung caa ma tran iecmit den thuc Tif suy mkti agh*n dac trung cila ma trail A a thuAn ao hac bang khong Gia this cat nghiem khac khong as da thud dee trung PA la: jai , , tat , -iak, (cti E K, x 0) Chi PA N= PA W + a ?) ) {0 vain >2k Do det A = PA (0) = k II ot i nefun=2k jo t ' 185 Do det A a TU day, de thAy detA = ngu n le Di& cling có the suy bang cluing mirth trkic tigp 3.15 Bo dg: Cho V la khong gian vec to tren truang C, U li anh xa nem tuygn tinh: V —> V, nghia la u (ax + py) = u (x)+ u (y) vol mgi x, y e V, a, p E C Khi u2 IA huh 3u tuygn tinh Gia sit X la mgt gia tri rieng thkic, am cua u , X la nghiem bOi than cim da thfic dac trung Put Chung mink be, dg: De thy u e End (V) Gin) six 11 mg gia tri rieng time < cim u va a x la vec td rieng cua u2 vdi u2 (a) = Aa Khi u(a) va a la dec lap tuygn tinh V That vay n'elk u(a) = -4 a, e C thi u2 (a) = u(4a) = u(a) = 141 a=X - Do X = 141 a 0, trai vdi X< O G9i W la khong gian vec td hai chigu cna V, sinh bai a u (a) De thAy moi vec td cim W den la vec to rieng vol gia tri rieng X va u (W) c W Dal V = V/W, xet anh xa can sinh u:Vc —> [x] —> u,[x] = [u(x)] Ta c6 u, IA anh xa nUa tuygn tinh, tit u 12 la anh xa tuygr tinh va u1 2[x] = [u 2(x)] VI vay, ki higu PIO va Pu, la cac dz thtc dac trung cliatt c End (V) va u12 c End (V1) ttiOng ling thi Pu2(t) = - A.)2 Pu 2(t) Niu y lai IA nghigm cf.a da thug 4( thing Pill ', lap lai qua trinh tren to có (t - )02 la ink Gila Pu l a s6 muQuatrinhy vd 186 na (t - A) phan b.& Put la s6 than B6 de' dude cluing ainh Chung mink bai Man: Xet u: C° —> C11 , u (x) = Al; u la Anh xa n&a tuyen tinh; t2 (x) = u(u00) = u (Al) = K Ax voi moi x e Ta co Pu2 (t) = let (A A — tIn) Viii moi t e R, ta en det (A.A -t In) = let (A A - t In) = det k -t In) (xem bat 2.51) Nhu \Tay Pu (t) a da dine vol he so' thve Pu2(t) =(%14)a' • 0.2 - 0'2 muyen during, ., Irk e R; Gin) - oak x dO sn ,sk Q e R Q khong co nghiem thne Vi Q khong có nghiem thile nen deg Q = n -(s, + s + + sk) Alan He se cao nhat cua Q(t) la (-1)"" sk) , nghia la bang Do Q(t) > vdi moi t e R, to Q(-1) > Bay gid ta xet the nghiem (i = 1, 2, , k) N6u CO < 0, thl boi s, ena nghiem ?9 chart, (Xi + nsi z N6u 7u z 0, thi re rang (X,+1) 31 >0.Nhv vay Pu2 (-1) a 0, nghia la det (A A+ In) a a , Chu $: Deu bang co th6 x637 ra, chamg han xet A = 3.17 Vol f e End (U) Neu f la t0 Tang caM tor lien h0p thi vdi moi x e U; ta co: < f(x), x >=== 187 Nhu vay thole hay thitc \TM moi x e U Node lai n6u moi x e U, to có thoc, to cheini minh f to lien hOp Than Lich f tting can hai to deing cM to lien hop: f = ft + i ft,, = + i Nhung ft, 1, la nhang to thing chin to lien hop nen