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1、一、高斯消元法以及（高斯消元法及矩阵）ThreePossibilities UNIQUESO1.UTION:TberViSOneandcmIyonee“valuw*forbeZJSchatsatfecsallequationssimultaneously. NOSO1.UTION:Then、运DOeofvaluforthrxlVthatatii!iHinmha!xi2ilytlwwlutioHetwempty. INFINITE1.YMANYSO1.UTIONS:TiMmart-hhndynuyiiflcrentHeUofVaJueeforthtiUlalSat耶allequationsxim
2、ultAiMMMudy.ItuIKMiifcdttoPrcyVrthatifwtemhasttnorvtluhton.tlrnithzinfinitelytnAnysolutioic*.R*rxAinplc,itwixnpus*iblcfuraxp*U11UjIiavcrxwtytwodiHcrrtt4uto!u.行阶梯阵及其秩RankofaMatrixSUPPoeCAmxnisreducedbyrowoperationstoanCChCkmformE.Tlrurilofisdefinedtobethenumberrank(=mmbrrOfPiVnU=numberofnonzeroToWxin
3、E=numberofbasiccolumnsinA.wlrvthebtmircolumnsofAarceculunmnhitlmttninthiu)dxiti(H!w*ingfeequivalenttosayingIIuUbUcotMhunt. Inrwreducing:AlIEa11,ftl,tmingf11neverpw3hnnilMwi!hnCoiiMiiiUKmofthe1nuq*luutin.(2.3.5)齐次系统A0系统为，假如有段有至少一个非零数字，即系统变成川以，此系统为.假如Av.,的秩是r，和基础列对应的&是，其它的是有r个，有-，个基础解是的线性组合当且仅当其秩为n时有唯
4、解Summan1.CtAmnbethecoefficientmatrixforaho11ogriMMnwMyMtrrnofmliiM)arCqiIa【iousinnUUknOWns.anduppobrank（八）=r. Theunknownsthatcorrespondtothepositionsoftheba*iccolumns(i.c.athePiVOulpce4tion)arccalledthebajtMmriablrtandtheUnknowi4OOrTeHPOndingtotheoe4tionofIbenonbaMcCUIUnUearcEHCdthenvmriablr, Then?a
5、reCxaCtlyrbajkvanabkal-rfreeVariBl)le. Todescribeallx!utkjuIredUCCAtoarowechelonformuingGatuvianelimination,andthenusebackMIIEitUtiontouohvforthebasicVariat4internbofIbefreeVariabk.ThibProdUCeHthegrtumlnolutumthatha*theformX=lh+rjh2-+Jiwhen?thetermsl.xjr/Rran*XlwfrwvariablesandWhCmIi1.h2bn.rarrnIcol
6、umn9thatrepresentPartieUhuBm1u-tkMsoflivehotnogciwousHystonuTIiollsarhkpngcn-Crat*tillpotuiblesolutions.AIxnog,n.aUniqUcrofunknowns.Tlicrcartnofreevariabk.cTheassociatedhomogeneousSyStelnpossessesonlythetrivialsolution.线性矩阵的应用一一基尔霍夫定律基本的(共枕转置)PropertiesoftheTransposeIfAandBarctwomatricesoftheSameblu
7、pe.andifiascalar,theneachofthefollowingstatementsistrue.(A+B)r-A+Brand(A-Bf-A,+B,.(3.2.1)(nA)7-0AranA.(3.2.2)满意条件/(x+y)=(x)+(y)fax)=af(x)的系统为线性系统.在平面几何中，线性系统指直线，在高阶系统可表示为多个变量的线性组合f(xl.x2.jcn)=a,.v1+a2x2+.+anxn两个矩阵的枳O代表了两个线性系统的，即两个矩阵的史合。DistributiveandAssociative1.awsFrWafSIuabkIiIalrim1ithek4kmiyfet
8、nc. A：BC)BAC(Irft-IiAiid!ix-OUnPbEthatx-O.t11lntivlaw). (D+E)F-DF*EF(血匕kdUw*iuPrUPettkbold.(A,),-A.(3.7.13)TbrproductABka!mnonMinulAr.(3.7.14).(3.7.15)(A,)三(Ar)1al(A,)-(A,.(3.7.16)三、两个矩阵之间的关系 相像矩阵时于矩阵A8当且仅当存在一个可逆方阵P使得产AP=B则称A相像于B,记为性质：秩相等、行列式相等、相等、相同的特征值，相同个数的特征多项式、相同的初等因子 等价矩阵对矩阵A3,当且仅当存在两个可逆方阵P.Q使
9、得IiAQ=B相像矩阵肯定是等价矩阵，反之不成立。等价矩阵其实就是矩阵经过初等行变换和初等列变换得到的矩阵，两者具有相同的秩 合同矩阵存在可逆矩阵P使得P,AP=B四、四个特别的向量空间SummanT1m*fofulM11mtalslxccsajciaUtu.R（八）AxRm.(Ar)三yCR.N（八）=xIAx=0)Cr(r)-yI,y=C Therangorcohmnspace: Trwmixorlfl-hndir: ThenulkqiiuC： Thlf-l*ul11u11jmu*:1.rtPbeanonsingularmatrixsw*hthatPA=U.whereUinrow*bk)n
10、fr11.11lzxx*rrank()t=r. Spumingsetfor（八）=tlIW4cCoIulnn5inA Spanningm?for(A)=thenonzerorowsinI： SjMUining*hfcr.V（八）=thl,inthe,nrnlMention(AxO. SanningsetforN(A,)=the1j4rn-rrowsofP.IfAandBhavethesameshape,th.V(A)=(B).五、线性相关或线性独立（）1.inearIndependenceandMatrices1.etAbeannIXnmatrix. Eachofthefallowingsta
11、tementsisequivalenttosayingthattheCOhHn4ofAformnlinearlyidrMiHl2V（八）三0.(4.3.2)trank（八）=n.(4.3.3) EachofthefollowingstatementsiscquiaalcnttosayingthattherowsofAforualinearlyindependentset.tN(A)0.(4.3.4)trank（八）三m.(4.3.5)WhenAisftSqHftrOmatrix,eachofthefollowingyitatTheColumnofAformalinearlyindependewtct.(4.3.6)tTlicrowsofA(innalinearlyinlciltt(4.3.7)线性空间的基（）张成这个空间所需的最小数目的线性不相关的向量C
