e0 ctRo DUcvA DAorAo Hq vd t€n thf sinh Sd bdo danh .{ DAI HOC HUE KV rHr ruydN srNHsAUDAr Hoc NAM 2006 Mdn thi: Dai sd (ddnh cho: Cao hgc) Thdi gian ldm bdi: 180phut Cflu 1. a. Cho n1,r2, .,r, id cr{cvectokhr{ckhOngctia m6t khOnggian vecto,cbn A le mQt ph6p bidn ddi tuydn tinh cfia khOnggian vectd d6 cho A r t - 1 ,A n p : t r k* r t - t , k :2,3, )n. Chtrngminh rang cdc vectdrr,n2, .,rn dOcldp tuyOntfnh. b. Cho B lil ma trAn vuOng cdp n cho Bk :0, vdi k le mQt sd tu nhiOn ndo d6. Tim (E^ - B)-t, d6 E- ld ma trAn vuOngdon vr cdp n. Chu 2. Cho c le nh6m sinh boi hai phdn tir r vd a vu cdc quan h€: tr4:a2:!,ar-r3a. a. X6c dinh nhfrng phdn tr?cfra nh6m G. b. Tim tdt ca ciic nh6m cfia nh6m G. CAu 3. Cho n le mQtvdnh. DFtt" Z(R) - {, e Rlra: an,Voe R}. a. Chtnrgminh rang Z(R) lh mOt vdnh giao ho6n cira R. b. Xdc dinh z(Mt(n)), d5 twt(n) lh vinh cdc ma trQnvuOngcdp h0 sd thuc. CAu 4. Chtnrg minh rang ndu da thrlc 13+ arz + br * c c6 nghiOmthuc, phAnbiet thi da thrlc 13+ ar2 + i@' + b)r + # cflng c6 nghigm th1rc,phAn bigt. Ghi chri: C6n b0 coi thi khOng giii thfch gi thOm f t e " t l '\)*- e0 ctAoDUcvA oao rAo Hg ud,t€,nth{ s'inh: Sd b6,odanh,: DAr HQCHU6 KV rnr ruyiN {.*?e srNHsAU o4l Hec NAnnzoor M6n thi: DAI SO (dd,nh, cho Cao hoc) TlLdi gian ld,mbd,i:180 phrit Cdu f. 1. Cho nrtfr2t .tnn Ib cri,cvectokhdckh6ngcrlam6t kh6nggian vecto vh,A th,mQt ph6p bi6n ddi tuydn tfnh cria kh6ng gian vectcrd6 cho A n : 1 , A n n : f r k* r n - r , k : 2,3r . rfl. Chfrng minh rH,ngc6c vecto nttn2r .,tfrn dgc 16,ptuy6n tfnh. 2. Cho B Ie ma tr$n vu6ng cdp n x6c dinh tr6n trudng F cho Bk:0, v6i k th mQt s6 tu nhi6n ni,o d6. Tim (E^ B)-1, d6 En lir, ma trAn vu6ng dsn vi cd,pn. tl a g . T f nor ( ( _, .12000 1\--0) / n --t-:(0 v6i ( _, 1\ L nxac x d cdinh d i n htran tr6n t . r b n sF.F . lb,ma trdn trulng 0) Ciu II. 1. Cho p ve ',hL hai tu dbng cdu tuydn tfnh cria mQt kh6ng gian vectcrhfru han chibu tr6n lrubng s6 phirc C chopo{s : tog. Chfrng -Ln rXng p ve 4t cd chung mQt vects ri€ng, 2. Cho E lb m6t kh6ng gian vectcrEuclid htru han chibu vh (u1, .,un) lb, m6t hO trgc chudntrong A.-CA*ng minh rXng n6u vcri ngi u e E tu dbu c6'-' rur'iro,,,r' 'i,:L thi (u1, ., or) lb mQt ccrs& a3,a E. C6.u IU. Cho G th mQt nh6m nh6,nhfru hg,osao cho G c6 mQt tu d8ng c6,ug th6a p(a) # o,Va t' 16. Chfrngminh rB,ng: 1. vcri m.oia e G tbn tq,i g e G saocho d': g-Lp(g); 2. ndu g c6 cdp bXng2,tftcIi, p+i,d,vdp2 - id, thi p(.q): g-L v6i moi g € G vb, G te mQt nh6m aben c6 cd,p14,m6t sd 16. C6.u IV. 1. Cho E le mQt vh,nhBiao hod,nv6i don vi I vA,/ lb mQt id€an crla E. Chfrng minhrXng v6i m5i a € R, t6,pcon J - {ar* I | * € l?} C RlI lh,m6t id6an cda Rl I sinh bdi o + f € R/1. Tri d6 suy rXng f ld,id6an t6i dai crla vA,nhR thi moi phb,ntrl kh6c kh6ng c,la RII dEu khd,nglrieh. 2. Chirng minh rXng t$p hqp c6c s6 hfru tj dang ? usimAu sd Ih,mQt s6 nguyOn16 n tao thb,nhm6t mibn nguy6n chfnh. Ghi chri: Cd,nb6 coi,th,i,kh,6nggi,di,th{ch gi,th€,m. e0 crAo DUcvA DAoTAo Hg ud t€n tht sinh: Sd bd,odanh: DAI HOC HUE Kv THr ruyiN srNH sAU D4r Hgc NAvr 2008 M6n thi: DAI 56 (ddnh cho Cao hq") Thdi g'ian ld"mbd,i: 180 phrit CAu I. o,. Ky hieu M"(F) lA,khong gian vecto cd,cma trAn vu6ng cdp n c6 he tri trudng F. Ma trAn A - (o"i). e Mn(F) gqi lb d6i xfrng ndu aai : aji, v6i moi i, j :1, .,D. Chirng minh rXng tap tdt cA c6cma trAn ddi xirng A/l.(F) lb mot khong gian crla M,(F). Tim mQt ccysd vi tfnh sd chibu cria khong gian cl6. b. Cho V vd14/ Id,hai khong gian vecto tr6n trudng /( vd W co sd chibu huu han. Cho f , V -+ W lb mot toi,n cdu tuy6n tinh. Chirng minh rHng tbn tai m6t dnh xa tuy6n tinh g : W ---+V cho f g - idp1,v6i idry ld 5nh xa dbng nhdt crja W. Anh xq g c6 nhdt khong? CAu II. a. Cht'rng minh rXng d6i vdi c6,cvecta r,, y, z bdt ky mqt khong gian vectcy Euclid, d8ng tht'rc sau 1u6n xAy ra: ( l " l ' + l a l 2+ l r l " )- l r * a * r l , + l r - a l ,+ l y- r l , + l , - r l t . b. C6 tbn tai hay khong hai ma tr5,n vu6ng A,B cdp n c6 he tri trulng tr voi A khA nghichsao choAB - BA: A? Cdu III. Ki hieu R le trudng cd,cs6 thuc. Cho: G-{(: t\o 1,)lo,b€R,o>o} "-r)ta,b€1r{,o) J b\,', l re} i.(, ;)lae vitH-{(t a. Chirng minh ring G le mOt nh6m vdi ph6p nhan ma trAn vd lI li mot nh6m chudn t6c cria G. Chi rXng nh6m thucrng G H d8ng cdu vdi lO"u I "f.Oc5c s6 thuc IR. b- Hay tim m6t nh6m chudn tXc K ciaG chira -tr c h o K + G v i t " K * U CAU IV. a' MOt mibn nguy6n ggi ld mibn nguyen chinh ndu moi idean crja n6 dbu 1A idean chinh. Chirng t6 ring vA,nh cdc s6 nguy6n 2,,Ib,mibn nguy6n chfnh nhung vdnh da thfrc V,ft] khong ph.Aili mibn nguy6n chfnh. b' Ky hieu Q["] Ib vdnh cdc da thirc co he tri trubng s6 huu ti e Cho p le, mQt s6 nguy6n td. Chirng minh rXng cla thirc t . t , t Ar(r) - 1* r t .* rP-T bdt khd quy trone Qlrl. Ghi chri: crin b6 coi,thi khong gid"i thtch gi th€m. Typeset by AlaS-Tlf, a0 crao DUCvA DAorAo Hg ua t€n tht s'inh: Sd bd,odanh: DAI }IOC HUE K:r rr{r ruy6N srNH sAU DAr Hgc xAnn 200e(Dqt r) M6n thi: O,A.f Sd (danh cho Cao hqr) Thdi g'ian ld,m bd"i,r180 phft CAU I. Cho U vh, W Ii,, hai kh6ng gian crl.alf-kh6ng gian vecto hfru han chi6u. {-'hr'rnsrnin',rr;\'rg c6c di6u kien sau 1},tucrng dUcrng: (f) Cir,.r{/* Cirn)Z - dinnr/ ; /i;\ rh. tai rngl tu tlbng c{"uf ciav cho U - I-/,W : Ker/. Ci"u II. 1. Cho C?ng to).n phucrng O tr€n khong gian vectcrthuc rz chibu 7'. Chring minh ring c5 thd khai trrdn V' thi,nh tdng truc tidp c6c kh6ng gian con: -----f) -*---tf' \ ,/.,-*t"7.^ V n : V o@ V +@ V trcng d5 1/3- K"rO, Ol rv+ x6c dinh d.ucrng, r e ' Ql lv- x6c dinh Am. FIc*nnfra, CimT-r-: sr dimV- - t, vdi s vh, t tucrng frng Id,chi s6 duong vd chi s6 6rn qu6n tfnh cda O. 2. i.I.{.ydua d?trg toin phuong sau tr6n kh6ng gian vectcrthuc R3 vE d+ng chfnh t6c: Q(r,a, z) :2r2 * r g - r z + a 2* z - 2 vi. xdc dinh co sd tucrng rlng. C6.u III. Coi Gtr(t, R) i), nh6m cdc ma tr5,n vu6ng thuc c6"pn khong suy bi6n v6i ph6p tod,nlir rrb6p nir6,nczicrna tr6.n, X C GL(n,R) It t6.p c6c rna tr6n c6 dinh thirc bing +1 vi Y C GL(rt.,lR) 1AtAp c6c ma trAn c6 dinh thirc ducrng. Chirng minh rXng: 1. X,Y 1),cdc nh6m chu6lnt6c cl,a GL(r,R); 2. nh5rn +"hucrng G[,(n,R)/X d8,ngcdu v6i nh6m nh6,nc6c s6 thuc ducrng; 3. r:irdm tirucrng GI-,,(n,R)/y di,ng cdu v6i mQt nh6m cyclic cdp hai. CAtr IV. Cho 1i i; mQt vl,nh. 1. Giti,su B kh6ng c6 udc cda kh6ng, c6 rngt phb,ntri rs l0 co cdp v6 han d6i vdi ph6o cQngvd, rnoi nh6m crLa(ft, +) vdnh dEu 1),id6an trd.i cria vdnh E. Ch'ing minh .ing vinh n d8,ng cdu v6i mQt v)"nh crj.avinh Z ci'"cs6 nguy6n. 2. Khi R : Zga, hdy chfrng minh di,ng cdu vb,nhsau: = Za. Rl$Z,196Z) . rAo .{ DAI HOC HUE Hq vd t€n thf sinh Sd bdo danh KV rHr ruydN srNH sAU DAr Hoc NAM 2006 Mdn thi: Dai sd (ddnh cho: Cao hgc) Thdi gian ldm bdi: 18 0 phut Cflu 1. a. Cho n1,r2,. danh,: KV rnr ruyiN srNH sAU o4l Hec NAnn zoor M6n thi: DAI SO (dd,nh, cho Cao hoc) TlLdi gian ld,m bd,i: 18 0 phrit Cdu f. 1. Cho nrtfr2t tnn Ib cri,c vecto khdc kh6ng crla m6t. tfnh cria kh6ng gian vectcr d6 sao cho An1 : 11 , Ann : frk * rn-r, k : 2,3r rfl. Chfrng minh rH,ng c6c vecto nttn2r ,tfrn dgc 16 ,p tuy6n tfnh. 2. Cho B Ie ma tr$n vu6ng