DANHSCHSINHVINNHM3LPTON2007B1TrnThTiNguyn2NguynThNgcHn3TrnThThuTrc4NguynLPhngThy5NguynVnNgha6HAnhThi7HKimChn8VHunhDuyKhnh9TrnVnhThng10NgVnTrng11NguynHinNhn0www.VNMATH.comMCLCPHNI-LTHUYT 11 nhxxnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Ccphpthuxtrongkhnggian Pn. . . . . . . . . . . . . . . . . . . . 43 Ccnhlcbncaphpbinixnh . . . . . . . . . . . . . . . . . . 7PHNII-BITP 9Tiliu 240www.VNMATH.comPHNIIBITP9www.VNMATH.comBi1:trang74Chokhnggianxnh(Pn, p, Vn+1).Ccphpvtca Vn+1idinchonhngphpbinixnhnocaPn?GiiCcphpvtk.idv: Vn+1Vn+1x k.idv(x ) = k.x (k K) l nh x tuyn tnh idinchophpbinixnhngnht.idv: Pn PnM idv(M) = MBi2:trang74Nubini xnhf:PnPngibtngr+1imclpnmtrnmtr-phngthncgibtngmiimcar-phngkhng?GiiNufgibtngr+1imclpnmtrnr-phngthfkhnggibtngmiimcar-phng.Chnghn,trong P2taxtphpbinixnhfcbiuthcta:___kx

0= 4x0 x1kx

1= 6x0 3x1kx

2= x0 x1 x2chongthng(d):x0 x1= 0.TathyfbinimA(0:0:1)vB(1:1:0)nmtrndthnhchnhn.NhngM(1:1:1)thucdmf(M) = (3 : 3 : 1) = M.Bi3trang74Trong Pnchor-phngU,trnUlyr + 2imtrongbtkR + 1imnouclp. Chngtrng, nur + 2imubt quaphpbini xnhca Pn, th miimcaUubtng.GiiGisS0, S1, . . . , Sr, Eln + 2imtrongr-phngUthaiukinbiton.Gi e0,e1, . . . ,er,e lnltlccvectidinchoS0, S1, . . . , Sr, Eln + 2.TrnUtacrimclpS0, S1, . . . , Sr, Eln + 2.VE Unntace = t0e0+t1e1+. . . +trer(1)te

0=t0e0,e

1=t1e1, . . . ,e

r=trer. ycnglccvecti inchoimS0, S1, . . . , Sr.(1) e =e

0+e

1+. . . +e

r10www.VNMATH.comVS0, S1, . . . , Sr, Ebtngnn(e

0) = k0e

0ki = 0, i = 1, r(e

1) = k1e

1k = 0. . .(e

r) = kre

rDo( e ) =e

0+e

1+. . . +e

r) k(e

0+e

1+. . . +e

r) = k0e

0+k1e

1+. . . +kre

r (k k0)e

0+ (k k1)e

1+. . . + (k kr)e

r= 0V {e

0,e

1, . . . ,e

r}clptuyntnhnnk = k0= k1= . . . = kr.LyM Ucvectidinl m.Khim= l0e0+l1e1+. . . +lrer (m) = kl0e

0+kl1e

1+. . . +klre

r)= k(l0e

0+l1e

1+. . . +lre

r= kmDof(M) = M.VymiimthucUubtng.Bi4:trang74TrongPnchophpbinixnhcbiuthcta:k.x

= A.x.Tmtaca:anhcasiuphngu = (u0: u1: ... : un).bTonhcaimX,= (x,0: x,1: ... : x,n).cTonhcasiuphngu,= (u,0: u,1: ... : u,n).Giia.Chosiuphngu = [u0: u1: ... : un].Tac:[u]t.[x] = 0.Mtkhcphpbinixnhbinsiuphnguthnhsiuphngu,cbiuthctalk.x

= A.x.Khitacaccimthucsiuphngu

tha:_[u]t.[x] = 0k.[x,] = A[x]_[u]t.[x] = 0k.[u].A1[x,] = [u]t.[x]DodetA = 0 = A1= k.[u]t.A1[x,] = 0 (1)11www.VNMATH.comDo(1)lnhcasiuphngu.b.ViX

= (x

0: x

1: ... : x

n)Suyratatonhl[x] = k.A1.[x

]c.Tac:u

= [u

0: u

1: ... : u

n] = [u

]t.[x

] = 0Khitaccimthuctonhcasiuphngu

tha_[u

].[x

] = 0k.[x

] = A[x][u

]t.A.[x] = 0Bi5:trang74Trong Pnchophpbini xnhf cbiuthc ta: k.x

=A.x. Gi () =det(A In)lathcctrngcamatrnA(Inlmatrnnvcpn).Chngminhrng:aTa (x0: x1: ... : xn) ca im bt ng l nghim ca h phng trnh: AIn)x = 0,tronglnghimcaathcctrng.bTa(u0: u1: ... : un)casiuphngbtnglnghimcah:(AtIn) = 0,tronglnghimcaathcctrng.cNulnghimncaathcctrngthimbtngvsiuphngbtngngvinghimkhngthucnhau.Giia)Phpbinixnhfcphngtrnh:kx

= AxKhiphngtrnhtmimbtngcafl:kx = Ax (A kIn)x = 0 (1)y l h phng trnh tuyn tnh thun nht c n + 1 phng trnh vi n + 1 n s. Munhphngtrnhnycnghimkhngtmthngth |AkIn| = 0viminghimk = ica h phng trnh |AkIn|. Ta c cc im bt ng ca f ng vi gi tr ring ic tathamnhphngtrnh(A kIn)x = 0.Bi6:trang75Trong P2chomctiuxnhS0, S1, S2; E.Tmbiuthctacaccphpbinixnhthamnmttrongcciukinsauy:a. CcSiulimbtng(tclbinthnhchnhn).b. CcimS0, S1, S2lnltbinthnhS1, S2, S0vimEbtng.c. imS0btng,ngthngS1S2btng(ngthngbinthnhchnhn)vS1binthnhS2.12www.VNMATH.comGiiGi e0,e1,e2lnltlccvctidinchoccimS0, S1, S2.a.vf(S0) = S0 (e0) = k0e0f(S1) = S1 (e1) = k1e1f(S2) = S2 (e2) = k2e2Biuthctacaphpbinixnhfl:___k

x

0= k0x0k

x

1= k1x1(k0; k1; k2 = 0)k

x

2= k0x2b.Vf(S0) = S1 (e0) = k0e1f(S1) = S2 (e1) = k1e2f(S2) = S0 (e2) = k2e0Biuthctacafcdng___k

x

0= k2x2k

x

1=k0x0(k0; k1; k2 = 0)k

x

2= k1x1Vf(E)=Enn___k = k2k = k0k = k1 k0= k1= k2= kChnk=1.Vybiuthctacafcntmcdng:___k

x

0= x2k

x

1= x0k

x

2= x1c.Vf(S0) = S0 (e0) = k0e0f(S1) = S2 (e1) = k1e2Biuthctacufcdng:___k

x

0= k0x0+a2x2k

x

1= k1x1 +b2x2k

x

2= c2x2PhngtrnhtngqutcangthngS1S2: x0= 0VM(0 : y1: y2) S1S2binthnhM

(0 : y

1: y

2) S1S2Nn___0 = +a2y2ky

1= k1y1 +b2y2ky

2= c2y2_a2= 0k1, c2 = 0Vybiuthctacafcntml:___k

x

0= k0x0k

x

1= k1x1 +b2x2k

x

2= c2x2vik0, k1, c2 = 0.Bi7:trang75Gi S0, S1, S2, S3; ElmctiuxnhtrongP3. Tmbiuthctacatt cccphpbinixnhf: P3P3thamnmttrongcciukinsauy:aCcimS0, S1, S2, S3iubinthnhchnhn.13www.VNMATH.combHaingthngS0S1vS2S3iubinthnhchnhn.cCcngthngS0S1binthnhngthngS2S3.dChchaiimbtnglS0, S2vchcmtngthngbtnglS1, S3.eChaingthngbtnglS0S1vS2S3,khngcimbtngvngthngbtng.f CcimS0, S1vmiimtrnS2S3ubtng.gCcimcamtphng< S0, S1, S2>ubtngvngthngS0S3btng.Giia.Biuthctacafcdng:k____x

0x

1x

2x

3____=____a 0 0 00 b 0 00 0 c 00 0 0 d________x0x1x2x3____Tac:f(E) = E

(f) :___kx

0= k0x0kx

1= k1x1kx

2= k2x2kx

3= k3x3Viki = 0ty,i = 0, 3.b.Gi e0,e1,e2,e3lnltlccvectoridincaS0, S1, S2, S3.Phngtrnhtngqutcaccngthng:S0S1:_x2= 0x3= 0S2S3:_x0= 0x1= 0Biuthctacafcdng:f:___k

x

0= a0x0 +a1x1 +a2x2 +a3x3k

x

1= b0x0 +b1x1 +b2x2 +b3x3k

x

2= c0x0 +c1x1 +c2x2 +c3x3k

x

3= d0x0 +d1x1 +d2x2 +d3x3VM(y0: y1: 0 : 0) S0S1binthnhM

(y0

: y1

: 0 : 0) S0S1.Nn:f:___ky

0= a0y0 +a1y1ky

1= b0y0 +b1y10 = c0y0 +c1y10 = d0y0 +d1y1___c0= c1= 0d0= d1= 0a0a1b0b1= 0TngtN(0 : 0 : z2: z3) S2S3binthnhN

(0 : 0 : z2

: z3

) S2S3.Nn:f:___0 = a2z2 +a3z30 = b2z2 +b3z3kz

2= c2z2 +c3z3kz

3= d2z2 +d3z3___a2= a3= 0b2= b3= 0c2c3d2d3= 014www.VNMATH.comVybiuthctacafcntml:f:___k

x

0= a0x0 +a1x1k

x

1= b0x0 +b1x1k

x

2= c2x2 +c3x3k

x

3= d2x2 +d3x3Vi___a0a1b0b1= 0c2c3d2d3= 0c.Phngtrnhtngqutcaccngthng(S0S1):_x2= 0x3= 0; (S2S3):_x0= 0x1= 0Biuthctacafcdng:f:___k

x

0= a0x0 +a1x1 +a2x2 +a3x3k

x

1= b0x0 +b1x1 +b2x2 +b3x3k

x

2= c0x0 +c1x1 +c2x2 +c3x3k

x

3= d0x0 +d1x1 +d2x2 +d3x3VM(y0: y1: 0 : 0) S0S1binthnhimN(0 : 0 : z2: z3) S2S3.Nn:f:___0 = a0y0 +a1y10 = b0y0 +b1y1kz2= c0y0 +c1y1kz3= d0y0 +d1y1___a0= a1= 0b0= b1= 0c0c1d0d1= 0Vybiuthctacafcntml:f:___k

x

0= a2x2 +a3x3k

x

0= b2x2 +b3x3kz2= c0x0 +c1x1 +c2x2 +c3x3kz3= d0x0 +d1x1 +d2x2 +d3x3___a2a3b2b3= 0c0c1d0d1= 0d.Tacf(S0)=S0,f(S2)=S2.Nn(e0)=k0e0 , (e2)=k2e2 .Biuthctacaphpbinixnhfcdng:___k

x

0= k0x0 +a1x1 + a3x3k

x

1= b1x1 + b3x3k

x

2= c1x1 +k2x+c3x3k

x

3= d1x1 + d3xPhngtrnhtngqutcangthng(S1S3):_x0= 0x2= 0VM(0 : y1: 0 : y3) S1S3binthnhM

(0 : y

1: 0 : y

3) S1S3.Nn:f:___0 = a1y1 +a3y3ky

1= b1y1 +b3y30 = c1y1 +c3y3ky

3= d1y1 +d3y3___a1= a3= 0c1= c3= 0b1b3d1d3= 015www.VNMATH.comVybiuthctacafcntml:___k

x

0= k0x0k

x

1= b1x1 + b3x3k

x

2= k2x2k

x

3= d1x1 + d3x3Vi___k0, k2 = 0b1b3d1d3= 0e.Theocubtacbiuthctacafl:f:___k

x

0= a0x0 +a1x1k

x

1= b0x0 +b1x1k

x

2= c2x2 +c3x3k

x

3= d2x2 +d3x3,vi___a0a1b0b1= 0c2c3d2d3= 0Tacphngtrnhcafl:a0 k a10 0b0b1 k 0 00 0 c2 k c30 0 d2d3 k= 0a0 k a1b0b1 kc2 k c3d2d3 k= 0 [(a0 k)(b1 k) b0a1][(c2 k)(d3 k) c3d2] = 0 (k2(a0 +b1)k +c2d3 c3d2) = 0_k2(a0 +b1)k +a0b1 a1b0= 0(1)k2(c2 +d3)k +c2d3 c3d2= 0(2) qua php bin i x nh f trong P3khng c im bt ng v mt phng bt ng thphngtrnh(1)v(2)chcnghimkp.Tcl_(a0 +b1)24(a0b1 a1b0) = 0(c2 +d3)24(c2d3 c3d2) = 0_(a0 b1)2+ 4a1b0= 0(c2 d3)2+ 4c3d2= 0___a1b0< 0c3d2< 0a0 b1= 2_a1b0c2 d3= 2_c3d2Vy16www.VNMATH.comf:___k

x

0= a0x0 +a1x1k

x

1= b0x0 +b1x1k

x

2= c2x2 +c3x3k

x

3= d2x2 +d3x3,vi___a1b0< 0c3d2< 0a0 b1= 2_a1b0c2 d3= 2_c3d2a0a1b0b1= 0c2c3d2d3= 0f.Theobitac:f(S0) = S0,f(S1) = S1f(S2) = S2,f(S3) = S3Khi:(e0) = k0e0,(e1) = k1e1(e2) = k2e2,(e3) = k3 e3Biuthctacafcdng:___k

x

0= k0x0k

x

1= k1x1k

x

2= k2x2k

x

3= k3x3PhngtrnhthamscangthngS2S3:_x0= 0x1= 0VM(0 : 0 : 1 : a) S2S3binthnhchnhnnntac:f(M) = M___k.0 = k0.0k.0 = k1.0k= k2ak= ak3 k2= k3= kTacbiuthctacafcntml:___k

x

0= k0x0k

x

1= k1x1k

x

2= kx2k

x

3= kx3(k0, k1 = 0)(k = 0)g.Tac:f(S0) = S0,f(S1) = S1,f(S2) = S2.Biuthctacafcdng:___k

x

0= k0x0+a3x3k

x

1= k1x1+b3x3k

x

2= k2x2+c3x3k

x

3= d3x3Phngtrnhtngqutca< S0, S1, S2>l:x3= 0.S0S3:_x1= 0x2= 017www.VNMATH.comVM(a : b : 1 : 0) < S0, S1, S2>binthnhchnhnnn:f(M) = M___ak = ak0bk = bk1k = k0k = 0d3 k0= k1= k2= kVN(1 : 0 : 0 : c) S0S3binthnhimN

(1 : 0 : 0 : c

) S0S3.Nn:___k = k0+a3c0 = b3c0 = c3ckc

= d3c___b3= 0c3=0d3 =0Vybiuthctacafcntml:(f) :___k

x

0= kx0+a3x3k

x

1= kx1+b3x3k

x

2= kx2+c3x3k

x

3= d3x3(d3 = 0, k = 0)Bi8:Trang76Cc php bin i x nh di y ca P3sinh ra nhng php afin no, gii thch nghahnhhccanhngphpafin?akx

0= x0, kx

i= xi, i = 1, 2, 3bkx

0= x0, kx

1= x1, kx

2= x2, kx

3= x3ckx

0= x0, kx

1= x1, kx

2= x2, kx

3= x3dkx

0= x0, kx

1= x1, kx

2= x2, kx

3= 2x1 2x2 x3Giia.Phpafinsinhrabiphpxnhl:X

i= Xi, i = 1, 2, 3nghahnhhc:Phpbiniafinsinhratrongnhxxnhtrnlphpixngtmlgcta.b.Phpafinsinhrabiphpxnhl:f:___X

1= X1X

2= X2X

3= X318www.VNMATH.comnghahnhhc:M(X1, X2, X3) f(M) = (X1, X2, X3)Phpafinsinhratrongnhxxnhtrnlphplyixngquamtphngx2= 0.c.Phpafinsinhrabiphpxnhl:f:___X

1= X1X

2= X2X

3= X3nghahnhhc:M(X1, X2, X3) A3 f(M) = (X1, X2, X3)Phpafinsinhratrongnhxxnhtrnlphplyixngquamtphngx3= 0.d.Phpafinsinhrabiphpxnhl:f:___X

1= X1X

2= X2X

3= 2X1 2X2 X3Bi1trang85:TrongP3chomctiu {S0, S1, S2, S3; E}.Vitbiuthccaphpthux1-cpvicslcpngthngS0S1,S2S3vctsk.GiiPhngtrnhtngqutcaccngthng(S0S1):_x2= 0x3= 0; (S2S3):_x0= 0x1= 0A(1 : 1 : 0 : 0) S0S1B(0 : 0 : 1 : 1) S2S3(AB):_x0 x1= 0x2 x3= 0lyE(1 : 1 : 1 : 1) AB.GiE

= f(E) AB E

(x0: x0: x1: x1) (x0, x1 = 0)Theogithittac:[E, E

, A, B] = k

1 11 0

1 01 1

:

x01x10

x00x11

= k x0= kx1VyE

(k : k: 1 : 1).Vflphpthux1-cpvicslcpngthngS0S1, S2S3nntac:f(S0) = S0, f(S1) = S1, f(S2) = S2, f(S3) = S3Biuthctafcdng:___k

x

0= a0x0k

x

1= b1x1k

x

2= c2x2k

x

3= d3x3Vf(E) = E

nn:___k0k = a0k0k = b1k0= c2k0= d3_a0= b1= ak2= d3= c19www.VNMATH.comVybiuthctacntmcafl:(f)___k

x

0= ax0k

x

1= ax1k

x

2= cx2k

x

3= dx3via,c =0Bi2trang85:TrongP2,chophpbinixnh:___kx

0= 2x0 +x1 +x2kx

1= x0 + 2x1 +x2kx

2= x0 +x1 + 2x2.Chngtrnglmtphpthuxcp.Xcnhcsvtsthux.GiiTmccimbtngcaf.Tac:2 k 1 11 2 k 11 1 2 k=0 k3+ 6K29k + 4 = 0 _k1= 4k2= k3= 1Vik1= 4tach:___2x0 +x1 +x2= 0x0 2x1 +x2= 0x0 +x1 2x2= 0 x0= x1= x2 O(1 : 1 : 1)Vik2= k3= 1tach:___x0 +x1 +x2= 0x0 +x1 +x2= 0x0 +x1 +x2= 0 x0 +x1 +x2= 0 x0 +x1 +x2= 0 vO V Nnflphpthux0-cpvicsl0-cp(O,V)Tsthux:LyB(0:1:-1) V,M=2(O)+(B)=(2:3:1) OBf(M) = M

= (8 : 9 : 7) OBk = [M, M

, O, B]=

2 13 1

2 03 1

:

8 19 1

8 19 1

= 4Bi3:trang85TrongP3chomtphngVcphngtrnh:x0 + x1 + x2 + x3= 0.GiflphpthuxnccsV,ctmthux(1:0:0:0).Tmbiuthctacaftrongcctrnghpsauy:aTsthuxk=3.bfbinim(0:1:1:1)thnhim(3:1:1:1).Tmtsthux.cfctnhchtihp,nghalf2lphpngnht.Gii20www.VNMATH.coma)LyM(0 : 1 : 0 : 0)/ V vM = O = (OM) :_x2= 0x3= 0A = OM V= A(1 : 1 : 0 : 0).Gif(M) = M

M

OM= M

(t1: t2: 0 : 0).M[O, A, M, M

] = 31 00 11 t10 t2:1 01 11 t11 t2= 3 t1= 2t2M

(2 : 1 : 0 : 0)LyN(0:1:-1:0),P(0:0:1:-1),B(1:0:0:-1)thucV.Biuthctacafcdng:___kx

0= a0x0 +a1x1 +a2x2 +a3x3kx

1= b0x0 +b1x1 +b2x2 +b3x3kx

2= c0x0 +c1x1 +c2x2 +c3x3kx

3= d0x0 +d1x1 +d2x2 +d3x3+f(0 : 1 : 1 : 0) = (0 : 1 : 1 : 0) =___0 = a1 a2k1= b1 b2k1= c1 c20 = d1 d2+f(0 : 0 : 1 : 1) = (0 : 0 : 1 : 1) =___0 = a2 a30 = b2 b3k2= c2 c3k2= d2 d3+f(1 : 0 : 0 : 1) = (1 : 0 : 0 : 1) =___k3= a0 a30 = b0 b30 = c0 c3k3= d0 d3+f(1 : 0 : 0 : 0) = (1 : 0 : 0 : 0) =___k4= a00 = b00 = c00 = d0+f(0 : 1 : 0 : 0) = (2 : 1 : 0 : 0) =___2k5= a1k5= b10 = c10 = d1=_k1= k2= k3= k5k4= 3k1Vy(f) :___kx

0= 3k1x0 + 2k1x1 + 2k1x2 + 2k1x3kx

1= k1x1kx

2= k1x2kx

3= k1x321www.VNMATH.com.Hay(f) :___kx

0= 3x0 + 2x1 + 2x2 + 2x3kx

1= x1kx

2= x2kx

3= x3b)(V):x0 +x1 +x2 +x3= 0.LyA,B,CthucV:A(0:0:1:-1),B(1:-1:0:0),C(1:0:0:-1).Biuthctacafcdng:___kx

0= a0x0 +b0x1 +c0x2 +d0x3kx

1= a1x0 +b1x1 +c1x2 +d1x3kx

2= a2x0 +b2x1 +c2x2 +d2x3kx

3= a3x0 +b3x1 +c3x2 +d3x3+f(A) = A. Suyra:___0 = c0 d00 = c1 d1k = c2 d2k = c3 d3(I)+f(B) = B. Suyra:___k = a0 b0k = a1 b10 = a2 b20 = a3 b3(II)+f(C) = C. Suyra:___k = a0 d00 = a1 d10 = a2 d2k = a3 d3(III)Tac:D(0:1:1:1),D

(3 : 1 : 1 : 1)+f(D) = (D

). Suyra:___3k = b0 +c0 +d0k = b1 +c1 +d1k = b2 +c2 +d2k = b3 +c3 +d3(IV )Tcch(I),(II),(III),v(IV),tasuyra:___a0= 2k0b0= c0= d0= k0b1= k1c1= d1= a1= 0a2= b2= d2= 0a3= b3= d3= 0c2= k2d3= k22www.VNMATH.comThayvobiuthctacaftac:___kx

0= 2k1x0 +k1x1 +k1x2 +k1x3kx

1= k1x1kx

2= k1x2kx

3= k1x3Vy:(f) :___kx

0= 2x0 +x1 +x2 +x3kx

1= x1kx

2= x2kx

3= x3(OD):___x0= t1x1= t2x2= t2x3= t2GiE= OD V= E(3 : 1 : 1 : 1)=[D,D

,O,E] =0 11 00 31 1:3 11 03 31 1= 2Vyk=-2.c)fctnhchtihp.Tac:f(M) = M

, f(M

= M)[O, A, M, M

] = k [O, A, M

, M] = K 1[O, A, M, M

]= k= k2= 1 k = 1 =_k = 1k = 1(loai)Dofctnhchtihpsuyrak=-1.M(0 : 1 : 0 : 0)/ V vO = M(OM) :_x2= 0x3= 0MA = OM V= A(1 : 1 : 0 : 0).GiM

= f(M) = M

OM= M

(a : b : 0 : 0)Theogithittac:[O, A, M, M

] = 1

1 00 1

1 a0 b

:

1 01 1

1 a1 b

= 11b:1a+b= 1 ab= 2 a = 2bChnb = 1 = a = 2 = M

(2 : 1 : 0 : 0).23www.VNMATH.comLyN(0 : 1 : 1 : 0), P(0 : 0 : 1 : 1), B(1 : 0 : 0 : 1)thucV.Tac:+f(0 : 1 : 1 : 0) = (0 : 1 : 1 : 0) =___0 = a1 a2k1= b1 b2k1= c1 c20 = d1 d2+f(0 : 0 : 1 : 1) = (0 : 0 : 1 : 1) =___0 = a2 a30 = b2 b3k2= c2 c3k2= d2 d3+f(1 : 0 : 0 : 1) = (1 : 0 : 0 : 1) =___k3= a0 a30 = b0 b30 = c0 c3k3= d0 d3+f(1 : 0 : 0 : 0) = (1 : 0 : 0 : 0) =___k4= a00 = b00 = c00 = d0+f(0 : 1 : 0 : 0) = (2 : 1 : 0 : 0) =___2k5= a1k5= b10 = c10 = d1Tcchphngtrnhtrntasuyra:_k1= k2= k3= k5k4= k1Vy___kx

0= k1x0 2k1x1 2k1x2 2k1x3kx

1= k1x1kx

2= k1x2kx

3= k1x3Hay___kx

0= x0 2x1 2x2 2x3kx

1= x1kx

2= x2kx

3= x3Bi4:trang85TrongP2choccimA=(1: 1: 1), B=(0: 1: 2), C(1: 0: 3), D=(1: 2: 0), E=(3:0:2).Tmbiuthctacaphpbinixnhf: P2P2,bitrngfgibtngccimA,B,CvbinimDthnhimE.cphilphpthuxkhng?Gii24www.VNMATH.comBiuthctacafcdng:___k

x

0= a0x0 +a1x1 +a2x2k

x

1= b0x0 +b1x1 +b2x2k

x

2= c0x0 +c1x1 +c2x2V f(1 : 1 : 1) (1 : 1 : 1)nn___k1= a0 +a1 +a2(1)k1= b0 +b1 +b2(2)k1= c0 +c1 +c2(3)V f(0 : 1 : 2) (0 : 1 : 2)nn___0 = a1 + 2a2(4)k2= b1 + 2b2(5)2k2= c1 + 2c2(6)V f(1 : 0 : 3) (1 : 0 : 3)nn___k3= a0 + 3a2(7)0 = b0 + 3b2(8)3k3= c0 + 3c2(9)V f(1 : 2 : 0) (3 : 0 : 2)nn___3k4= a0 + 2a1(10)0 = b0 + 2b1(11)2k4= c0 + 2c2(12)(*)T(1),(4),(7)tac___a0=3k1 +k34a1=k1 k32a2=k3 k14T(2),(5),(8)tac___b0=34(k1 k2)b1=k1 +k22b2=k2 k14T(3),(6),(9)tac___c0=3k1 + 3k3 6k24c1=2k2 3k3 +k12c2=3k3 k1 + 2k24Thayccai, bi, ci(i = 0, 2)voh(*)tac___7k1 3k3= 127k1 +k2= 07k1 + 2k2 9k3= 8k4___k1= k4k2= 7k4k3= 53k3Do___a0=13k4a1=43k4a3= 23k4,___b0= 6k4b1= 3k4b3= 2k4,___c0= 10k4c1= 4k4c3= 5k4Vybiuthctacafcntml:25www.VNMATH.com(f):___k

x

0=13x0 +43x1 23x2k

x

1= 6x0 3x1 2x2k

x

2= 10x0 4x1 5x2k = 0Hay(f):___kx

0= x0 + 4x1 2x2kx

1= 18x0 9x1 6x2kx

2= 30x0 12x1 15x2k = 0Phngtrnhttrngcafl:1 k 4 218 9 k 630 12 15 k= 0 k323k227k + 315 = 0 __k1= 3k2= 21k3= 5V phng trnh t trng ch c nghim n nn f ch c 3 im bt ng l A, B, C.VABC=1 1 10 1 21 0 3= 4 = 0NnA,B, Ckhngthnghngdof khngphi lphpthuxv khngtntingthngdnomiimtrndquafubtng.Bi5trang86:TrongP3chohaingthngdvdlnltcphngtrnh:_x0 x1= 0.2x0 +x2 + 3x3= 0._4x0 3x1 +x3= 0.x0 +x2= 0.Tmbiuthctacaphpthux1-cpvicslcp(d,d)vtsthuxk=-1.Giit(d):_x0 x1= 0.2x0 +x2 + 3x3= 0.(d):_4x0 3x1 +x3= 0.x0 +x2= 0..TrndlyimA(-2:-2:1:1)vB(1:1:-2:0).TrndlyimC(-1:0:1:4)vD(0:1:0:3).LyG(3:3:0:2) dvH(1:1:-1:-1) d.Phngtrnhthamscangthng26www.VNMATH.com(GH) :___x0= 3t1 +t2x1= 3t1 +t2x2= t2x3= 2t1 t2PhngtrnhtngqutcaGHl:(GH) :_x0 x1= 0.2x1 + 5x2 3x3= 0.LyM(4:4:-1:1) GH.Khif(M) = M

(a : a : b :2a + 5b3) GH.Theogithitk = 1 [M, M

, G, H] = 14 31 04 31 0:a 3b 0a 1b 1= 1 a = 2bVyM(2:2:1:3).V f l php thu x 1 - cp vi cp c s l cp ng thng d v d nn f gi bt ng ccimA,B,C,D.Biuthctacafcdng:___kx

0= a0x0 +a1x1 +a2x2 +a3x3kx

1= b0x0 +b1x1 +b2x2 +b3x3kx

2= c0x0 +c1x1 +c2x2 +c3x3kx

3= d0x0 +d1x1 +d2x2 +d3x3TacA A___2k0= 2a0 2a1 +a2 +a3(1)2k0= 2b0 2b1 +b2 +b3(2)k0= 2c0 2c1 +c2 +c3(3)k0= 2d0 2d1 +d2 +d3(4)B B___k1= a0 +a1 2a2(5)k1= b0 +b1 2b2(6)2k1= c0 +c1 2c2(7)0 = d0 +d1 2d2(8)C C___k2= a0 +a2 + 4a3(9)0 = b0 +b2 + 4b3(10)k2= c0 +c2 + 4c3(11)4k2= d0 +d2 + 4d3(12)27www.VNMATH.comD D___0 = a1 + 3a3(13)k3= b1 + 3b3(14)0 = c1 + 3c3(15)3k3= d1 + 3d3(16)M M___2k4= 4a0 + 4a1 a2 +a32k4= 4b0 + 4b1 b2 +b3k4= 4c0 + 4c1 c2 +c33k4= 4d0 + 4d1 d2 +d3()T(1),(5),(9),(13)tac:___2k0= 2a0 2a1 +a2 +a3k1= a0 +a1 2a2k2= a0 +a2 + 4a30 = a1 + 3a3___a0= 2k0 + 3k1 4k2a1= 3k1 + 3k2a2= 2k0 k1 k2a3= k1 k2T(2),(6),(10),(14)tac___2k0= 2b0 2b1 +b2 +b3k1= b0 +b1 2b20 = b0 +b2 + 4b3k3= b1 + 3b3___b0= 2k0 + 3k1 5k3b1= 3k1 + 4k3b2= 2k0 k1 k3b3= k1 k3T(3),(7),(11),(15)tac___k0= 2c0 2c1 +c2 +c32k1= c0 +c1 2c2k2= c0 +c2 + 4c30 = c1 + 3c3___c0= k0 6k1 + 4k2c1= 6k1 3k2c2= k0 + 2k1 +k2c3= 2k1 +k2.28www.VNMATH.comT(4),(8),(12),(16)tac:___k0= 2d0 2d1 +d2 +d30 = d0 +d1 2d24k2= d0 +d2 + 4d33k3= d1 + 3d3___d0= k0 + 16k2 15k3d1= 12k2 + 4k3d2= k0 + 4k2 3k3d3= 4k2 3k3.Thayvo()tac___6k0 + 2k1 4k2= 2k46k0 + 2k1 4k2= 2k43k0 4k1 + 4k2= k43k0 + 16k2 12k3= 3k4___k0= 4k4k1=12k4k2=74k4k3=74k4Thayccgitrcak0, k1, k2, k3tmai, bi, ci, dii = 1, 3___a0=52k4a1=154k4a2=234k4a3= 54k4;___b0=34k4b1=112k4b2=234k4b3= 54k4___c0= 214k4c1=94k4c2= 54k4c3=34k4;___d0=94k4d1= 0d2= 54k4d3=74k4.Vybiuthctacafcntml29www.VNMATH.com___kx

0=52x0 +154x1 +234x2 54x3kx

1=34x0 +112x1 +234x2 54x3kx

2= 214x0 +94x1 54x2 +34x3kx

3=94x0 94x2 +74x3___k

x

0= 10x0 + 15x1 + 23x2 5x3k

x

1= 3x0 + 22x1 + 23x2 5x3k

x

2= 21x0 + 9x1 5x2 + 3x3k

x

3= 9x0 9x2 + 7x3Bi8:trang86Trong Pnvi mc tiu x nh {S0, S1, . . . , Sn; E} cho php thu x n fkhc php dngnhtsaochomiimSiubtng.aChngtrngtmcaphpthuxlmttrongccnhSi.bVitbiuthctacafnuSiltmthux.GiiVflphpthuxnnntrongn + 1imbtngSi,(i=0, n)phicnimthucsiuphngbtng.GiilsiuphngiquaccnhtatrnhSi.Ly M ikhi : M(1 : 1 : . . . 1 : 0 : 1 : . . . : 1). V cc im Sj, (i = j) v M u btngnnbiuthctacafcdng:_kx

i= aixikx

j= ajxj(i = j)VSicngbtngnntac:f(Si) = Si f:_kx

i= aixikx

j= axj(i = j)Trongai = avnuai= athflphpngnht.TaschngminhimSiltmcaphpthux:TacSibtng. Gi dlngthngbtk i quaSi. Taschngminhdlbtng.LyX(x0: x1: . . . : xn) df(X) = (ax0: ax1: . . . : aixi: . . . : axn)= aM+xi(ai a)Si dVySiltmthuxcaf30www.VNMATH.comTiliu1.VnNhCng-Hnhhcxnh-Nhxutbnihcsphm.2.PhmBnh-Bitphnhhcxnh-Nhxutbnihcsphm.3.NguynMngHy-Bitphnhhccaocp-NhxutbnGiodc.31www.VNMATH.com