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启航考研数学资料启航考研数学资料 承载理想 启航未来 2010?Zhang K.L.11.1?0|A|6=0(An)|A|=0(r(A)=r)1.A!#$%&!(2.r(A)=n3.A)*,+.-0/214356784.A)9:;05.Ax=0?AB x 6=0,Ax 6=06.Rn,Ax=CDEB7.A0AFGH8.AIJKE9.A=P1P2 Ps,PiLI10.MNnHB,OPAB=EQBA=E11.AR!#$%&!(1.A=#$!%&(2.r(A)8K=0)9:Y1436.A0AZFGH7.AIJK ErOOO!,r(A)=r8.|A|=0,(A:=#$!%&)(-*)GDn=|(aij)n|=?a11a12a1na21a22a2n.an1an2ann?=Pj1j2jn(1)(j1j2jn)a1j1a2j2anjn-*)6_1.1abcdaef abcghij_1.2kfabclmn(a)oabpgiqj_1.3abYrs4(a)ctYu4vwxyabczY|oYababclm(a)j_1.4abYr4(a)c0,n(a)koab0._1.5A=(aij)nY4oai1Ak1+ai2Ak2+ainAkn=(|A|(i=k)0(i 6=k);a1jA1k+a2jA2k+anjAnk=(|A|(j=k)0(j 6=k)rAijvb(1 i,j n)._1.6nnabD1,D2Yi(a)zokc(a)4Y4oD1+D2wnaboi(a)D1D2ci(a)ckpo(a)dD1(D2)451启航考研数学资料启航考研数学资料 承载理想 启航未来 _1.7abcYs4(a)cY4yY4(a)ckabcghi.1.1.1abYr4|A|=0ch.1.1.2|aE+bA|=0 r(aE+bA)1.2.9abcY()1.2.10gab()1.2.1()c.1.2.2cY4oab?y.1.2.3abcY4.143?IJ HIJ HS?H?=?6?6?6.22.1?:2.1.1 A1=A|A|abcd!1=1adbc dbca!2.1.2(A.E)(E.A1)2.1.3a1a2a31=1a11a21a3a1a2a31=1a31a21a12.1.42.4.2HGAnY,?B,?AB=BA=E,?Aw,B?!Y.?#HA=(Aij)0=A11A21An1A12A22An2.A1nA2nAnnorAij|A|r4cvb.2.2?$2.2.1 AmAn=Am+n(Am)n=(A)mn2.2.2A=0,r=(a1,a2,an)0,=(b1,b2,bn)0n%a&)(oAk+1=dkA,*?+d=0.2.2.1 A=0,A=a1b1a1b2a1bna12b1a2b2a2bnanb1anb2anbn,-?.A=0;d=0=053启航考研数学资料启航考研数学资料 承载理想 启航未来 2.2.3A=P11.nP,Ak=P1k1.knP2.2.2 AY|4Yr4cP11.nP/bo?0Y2143cj65?7?,yAcgd2&8(j8?mc*c?/bhwY89yc;:2.3?&?(1,2,n,Bca&?(1,2,sAB=Cms(1,2,n)b11b12b1sb21b22b2s.bn1bn2bns=(c1,c2,cs)Ai=cio(i=1,2,s)iAx=cic A(1,2,s)=(A1,A2,As)=(c1,c2,cs)c1,c2,csw21,2,n.?AC)-Y143?B2CB)-Y14356?D?EoA0W?FH.2.3.2 1)kYG?!Y,2H4ckY4c?I?J?!Y4c2&8(?K2)kY8!YL?!Y,2H4ckY4c?I?J?!Y4ca2&8(.2.3.3nY4kY8!Y42M4kY4ck2.4?N?O?2.4.1/?P?!Y4c?Q?R?!Y4 ABCD!0=A0C0B0D0!2.4.2/SPS!cS!AB!1=A1B1!AB!1=B1A1!ACOB!1=A1A1CB1OB!AOCB!1=A1OB1CA1B!2.4.3/SPkA=A11A22!,B=B11B22!AB=A11B11A22B22!2.4.4/?PkY4c?T?U?!Y4 AB!=BAAB!54启航考研数学资料启航考研数学资料 承载理想 启航未来 2.5?V2.5.1!Y4?Wc(|A|6=0)(I)AX=B(II)XA=B(I)c(A.B)(E.X)(II)cXbn?Y?Q?RA0X0=B0,(I)c2ZX0,?Q?R?X;2.5.2Ax=0dBx=04,+A,Ba2&8(YY/,8?_c?a?bY8ck,d?e?Kf8?_kc?/?ccY4?Kg8?_Y4c2h4Y?i.2.5.3!AmndBlnc&)(jcjkljJjWjcAx=0dBx=0 PA=B+mGw?!YP/nK!YAmndBlnca2&8(?c?k AQ=B+mLw?!YQ/.2.5.4o?p1,2,sAx=0c?q2i4c?r?s1,2,s5678?Kf1,2,stAx=0)B?Kgs=n r(A)u?vBY1432w2x8C8y?z3)?v.2.5.1?l?J?W?cc?q2i4h?.2.6?|2.6.1!c&)(jccje=a&)(jccje=!cje.jjj!cjecYc.2.6.2!Y4c?ifh?i?!Y4c?e,.h?ia2&8(24cY?i8K2.6.3!cajifhjij!cje,.hji&)(cji.!cjifh?i?!Y4c?e.2.6.4!Y4c?if?!Y4cY?i41)kA?J?if?yc?!Y,c?!Y8GAK2)kA?J?aif?yc?!Y,c?!Y8LA.2.6.5!Y4c?ec?(A 6=O r(A)10 r(Amn)min(m,n);fr(A)=r(A0)=r(A0A);gr(kA)=r(A),(k 6=0);r(A B)r(A)+r(B)maxr(A),r(B)r(A,B)r(A)+r(B);r AOOB!=r OABO!=r(A)+r(B)r ACOB!6=r(A)+r(B)r(AB)minr(A),r(B)Amn,Bns,r(AB)=0r(A)+r(B)nA,r(AB)=r(B);B,r(AB)=r(A)r(Amn)=n(Ax=0M4r(AB)=r(B).ASSGS(AB=O B=0AB=AC B=C55启航考研数学资料启航考研数学资料 承载理想 启航未来 r(Bns)=n r(AB)=r(B).B?!Y4Yr4?L24?.LI)G?!YE?J?if?yc?!Y8?Y(2).?H?!Y4w?2Z4?r?oc0K242M4oco2jc?|4c4.Hc1o.?*?Y4?ac0?o?28?!Y.A)&G!YAcY4vbc?24?Ac?e.4?!YAhcrvb,.lmr+1vb?,?!YAc?er.?r(A)=r2.7?_1)?-*|E(i,j)|=1|Ei(k)|=k|Ei,j(k)|=1?E(i,j)0=E(i,j)Ei(k)0=Ei(k)Ei,j(k)0=Ej,i(k)E(i,j)1=E(i,j)Ei(k)1=Ei(1k)Ei,j(k)1=Ei,j(k)?#HE(i,j)=E(i,j)Ei(k)=kEi(1k)Ei,j(k)=Ei,j(k)2)?F?-*?S?(A0)0=A(kA)0=kA0(A B)0=A0 B0(A1)0=(A0)1(AB)0=B0A0|A0|=|A|(A0)=(A)0(A1)1=A(kA)1=k1A1(A B)16=A1 B1(A1)k=(Ak)1=Ak(AB)1=B1A1?A1?=|A|1?#(A)=|A|n2A(kA)=kn1A(A B)6=A B(A1)=(A)1=A|A|(AB)=BA|A|=|A|n1(Ak)=(A)k3)?_r(A)=nr(A)=n1r(A)=n 10r(A)n 1;|AB|=|A|B|;|kA|=kn|A|;?Ak?=|A|k;|A B|6=|A|B|;AA=AA=|A|E(b?r?s?)33.1?_1.AY143?Y143?)56?,AY143?Y143F?.2.?vAY14356S8?K?v!AY1435678.56启航考研数学资料启航考研数学资料 承载理想 启航未来 3.?S8,?S8?K?78,?78.4.Y143?78,?Y143?78?K?Y143?S8,Y143?S8.5.?vY14356S8?28?K?F?)!AY143?5678.6.143?1,2,nw8EY143i(1 i n)t?Y143?)56?.7.13j?1,2,n56S813j?w)jE13C4Tjn1v1356jDjE8.143?1,2,n5678143?2w8uE?vY143it=?B2C4T?n 1vY14356?D?E.9.m*Y143?1,2,n56S8 r(A)A,BY143?IJ,X&SI)Y143?=EGIJ.2.HABY143?IJ r(1,2,n)=r(1,2,n)=r(1,2,n,1,2,n)HABIJ.3.143?1,2,s2C143?1,2,n56?D?E AX=BB r(1,2,n)=r(1,2,n,1,2,s)r(1,2,s)r(1,2,n)4.143?1,2,s2C143?1,2,n56?D?E,s n,1,2,s56S8.5.143?1,2,s5678,2C1,2,n56?D?E,s n.6.143?1,2,s2C143?1,2,n56?D?E,r(1,2,s)=r(1,2,n),?Y143?IJ7.EY143?)?78?IJ.143?)?v?78?IJ.8.143?)?78?=DE,?78?Y143?v?FDE?G.9.?v5678)Y143?IJ,?)Y143?v?FSI.10.Am nH,r(A)minm,n,r(A)=m,A)-Y1435678?K11.r(A)=n,A)*Y1435678,X1,2,n5678.57启航考研数学资料启航考研数学资料 承载理想 启航未来 12.1)2C1,2,n56?D?E Ax=B r(A)=r(A)n*Ax=7?BD?E?=?E 1,2,n56S8 Ax=0!AB=n*Ax=DE?BD?E?E 1,2,n5678 Ax=0?AB2)=2C1,2,n56?D?E Ax=7B r(A)6=r(A)r(A)m nH,r(A)=m,r(A)=r(A.)Ax=EGB,4)?m n?,EG=DEB?Wc&8(c?%A)E?v?.5.1.1 f()=n+(a11+a22+ann)n1+|A|5.1.2|E A|=0 Ax=x AxdxY.5.1.3Y4 Y4 kY4cg,kY4cn.5.2?_?f()HA)9:?o1,2,nHA)H0/%$A BABF?S?B=P1AP+PF?H,XPP0=E/A?&S?$A?S?.$A +mA)S?(?)/*?+5.3.1Aw?,kY4 n r(iE A)=kikiCN ic A-n?b4c2&8(.*?PAc2&8(?.c?!Y,P1APkY,kY4cAcg.ikic?bY4c2&8(,A(1,2,n)|zP=(A1,A2,An)=(11,nn)=(1,n)|zP1.n|z5.3.1Hi=0Acg?,Aw?,kY4 ic=n r(A)=Ax=0?qi4c.5.3.2Aw?,kY4,4gc,+.?/0/=r(A).5.3.3n?!YAne?0cg,Aw?,kY4.5.3.4A Ak=PkP1,(A)=P()P1=P(1)(2).(n)P15.3.5?,?!Y4ctrA=trBf|A|=|B|dA,B8?whwgr(A)=r(B)A0 B0KA1 B1+.A,Bw0/nKA B60启航考研数学资料启航考研数学资料 承载理想 启航未来 Ak Bk+k2140/nKf(A)f(B),|f(A)|=|f(B)|A B,C D AC!BD!|E A|=|E B|,dA,BY4cg,2&8(hY.5.3.2 xA40c2&8(,P1xB40c2&8(.5.3.6?(?!Y?M4d43?5?,.5.3.7k?!Y4cg264,2&8(26?&8(?KfhY4gkc2&8(?7?8?Kk?9?:Y,hY4gkc2&8(?bY8Kgw?7?8?!Y4?,kY4,4l26;?Jw?7?8if?Y,4l26;?Jw?wif?2?p?f?I?2?R2?d?M.O?P6.2.2tV?T2U2?p?fA B61启航考研数学资料启航考研数学资料 承载理想 启航未来 O?P6.2.3tV?T2U2?p?fr(A)=r(B)O?P6.2.4f(x1,x2,xn)=x0Ax*R?S?2?G?d?x=CyQf=nP1diy2ia?b?.O?P6.2.5?abp_?,?RSI,M?VMpr(A)(lm?R2?d?M?)?_?.O?P6.2.6a?b?2?2MdiQ-1n0n1,Q2.O?P6.2.7?T2U?RYX2N%?d?M?l?m?RYX2N%A?C?K?V?M.O?P6.2.8(?)_TUA_U1.11.10.02.6.3?6.3.1?Q?a?b?6.3.2?R?S?Q?a?b?:1)2A?A?C?K?A?C2DFh2)nV?A?C2DF?R?S?z?h3)CXRSTUN,x=Cy,q(Cy)0A(Cy)=y0C0ACY=y1C0ACY=(y1,y2,yn)d1d2.dny1y2.yn?Qf=nP1diy2i,?2?diQA?A?C?K.?6.3.3(?)1,2,3?d?2,1)?1=12=2(021)(011)13=3(031)(011)1(032)(022)22)?f1=1k1k2=2k2k3=3k3k3)?f62启航考研数学资料启航考研数学资料 承载理想 启航未来?,?.?fx1+x2 x3=0?1=(1,2,1)0,2=(1,0,1)0.6.4?x1,x2,xn?,f(x1,x2,xn)0.?.O?P6.4.1f(x)=x0AxQ?R?Xu?_?4N%f1)x 6=0,x0Ax 0h2)f?nh3)A?0h4)A?!?#?$?%?0h5)AE?,?&?C(?)C0AC=Eh6)&?C,(?)C0AC=C1AC=12.nXi?0N.7)&?P,(?)A=P0PhO?P6.4.22?*?+?R?d.O?P6.4.3 AQ?R?T2U,+?faii 0h|A|0.O?P6.4.4-AQ?R?T2U A0,A1,Arp?R?T2U.O?P6.4.5-A,BQ?R?T2U,qA+BQ?R?T2U,AB,BA?_?Q?R?T2U./?n0?1?2?3?4(1)1?2?3?45V687n98:,;?AVB?C,D?EV:,;F?G?H8:,;?IG?J?K?L?M?N?OAP?V8:,;?C?Q?R?SUTF?G?HIN?O?V6h,V,k RP+V,k V(2)W?3?4X5V,WY?8:,;?C?Q?RAW VPW6V?$C?Q?R(3)Z?0?X5V8:,;?C?Q?AVr8:,;1,2,r?_(i)1,2,r?a?bh(ii)V,c?d:,;?e?871,2,r?f?g.PBh:i;kj1,2,rh:i;kCkQVkd?(?kl)RrBh:i;kCkQV9,mknV?r98:,;?C?Q?R?o?p?RA8:,;?C?QVq?APV90R098:,;?C?r?s?t?d?8:,;?R(4)u5v:w;?j1,2,nn9v:w;?C?QV?d?V.=x11+x22+xmmP?x1,x2,xn?1,2,nx?y?z.(5)1?2?3?4Rn?n98:,;Rn6dn98:,;?C?Q?|,c?n?abn98:,;?=?:,;?j?e?6?d?RA?o?p?n9?y?z:,;1=(1,0,0,0),2=(0,1,0,0),4=(0,0,0,1)63启航考研数学资料启航考研数学资料 承载理想 启航未来 =:;j6:;CQRnd?R?1,2,nx?cdn9:;=(x1,x2,xn)Rn?y?z?6n?;?fx1,x2,xn.?6:Rn|?dR?Z?u?(1)Z?X51,2,nH1,2,n6n98:,;?C?QVJ?AE1=c111+c122+c1nn2=c211+c222+c2nnn=cn11+cn22+cnnn(1)?f?g?12.n=c11c12c1nc21c22c2ncn1cn2cnn12.n=P012.n(1,2,n)=(1,2,n)P(2)kP=(cij)n(1)(2)%B?kkkk%,BPBh7?1,2,n?1,2,n?R(2)u5Vn9:;CQ?V?1,2,nxyz(x1,x2,xn),?1,2,nx?y?z(y1,y2,yn),J?_b?%(2),P?y?z?%(x1,x2,xn)0=P(y1,y2,yn)0?(y1,y2,yn)0=P1(x1,x2,xn)0?1?2?(1)125n9:;=(a1,a2,an)0,=(b1,b2,0bn)0,P(,)=0=a1b1+a2b2+anbn8:,;8,R?o?p?A(,)=0n?8:,;?R(2)1?2?|=0=p(,)n98:,;?()R(3)?1?2?g51,2,rdj?B?n98:,;?A?J?J?,P?1,2,rd?:,;?j?Rik?:i;kjkn9h:i;1,2,r6djJkJk?Bkh:i;kRP1,2,r?a?b?AE?r nn?A?n rn98:,;?A(n8:,;J?J?R(4)?g?f(,)0,(,)=0 =0i?f(,)=(,)j,f(,1+2)=(,1)+(,2)(1+2,)=(1,)+(2,)(c,)=c(,)=(,c)Zhang K.L.201064