线性变换的秩与其对应矩阵的秩的等价性

2009年3月第23卷第1期YINSHANACADEMICJOURNALMar.2009Vo1.23No.11,2(1.,010435;2.,014030):线性变换是一个抽象的概念,求一个线性变换的秩是一个难点,而每一个线性变换在一组基下都对应一个矩阵,矩阵的秩容易求.本文给出了线性变换的秩与其对应矩阵的秩的关系及其应用,从而使抽象的问题具体化:线性变换;线性变换对应矩阵;秩:O15121:A:1004-1869(2009)01-0023-02:nV,1,2,,nV,(V)(1),(2),,(n),(V)=L((1),(2),(n))dim(V)=R((1),(2),(n))1nV1,2,,nV,A,=A,R()=R(A):R()=dim(V)=R((1),(2),,(n)),!R((1),(2),,(n))=R(A),(1,2,,n)=((1),(2),,(n))=(1,2,,n)A∀∀R(A)=r,Ar!1,!2,,!r,xi(i=1,2,,r)#Px1(1)+x2(2)++xr(r)=((1),(2),,(4))(x1,x2,,xr)T=(1,2,,r)(!1,!2,,!r)(x1,x2,,xr)T=01,2,,r!1,!2,,!r,!x1=x2=A=xr=0!(1),(2),,(r)xi(i=1,2,,r+1)#Px1(1)+x2(1)++xr+1(r+1)=(1,2,,r+1)(!1,!2,,!r+1)(x1,x2,,xr+1)T=0,!1,!2,,!r+1!x1,x2,,xr+1!(1),(2),(r+1)(r+1),R((1),(2),,(r+1)=r=R(A)∀∃dim(V)=r,(1),(2),,(n)n,xi(i=1,2,,r)#P,x1!1+x2!2++xr!r=(!1,!2,,!r)(x1,x2,,xr)T=0,(1),(2),,(r)(!1,!2,,!r)(x1,x2,,xr)T=x1(1)+x2(2)++xr(r)=0(1),(2),,(r),!x1=x2==xr=0,!1,!2,,!rxi(i=1,2,,r+1)#P,x1!1+x2!2++xr!r+xr+1!r+1=0x1(1)+x2(2)++xr(r)+xr+1(r+1)=0(1),(2),,(r+1),!x1,x2,,xr+1,23:2008-11-28:丁万龙(1981-),男,内蒙古通辽人,研究方向:应用数学!!1,!2,,!r+1R(A)=r=R((1),(2),,(n))21:VP2,V,(x)=1-1-11x,!x#V:R()=dim(V)=R((1),(2),,(n))=R(A)VE1,1=1000,E1,2=0100,E2,1=0010,E2,2=0000,E1,1=AE1,1=1-1-111000=10-10=(E1,1,E1,2,E2,1,E2,2)10-10,E1,2=(E1,1,E1,2,E2,1,E2,2)010-1,E2,1=(E1,1,E1,2,E2,1,E2,2)-1010,E2,2=(E1,1,E1,2,E2,1,E2,2)0-101,!E1,1,E1,2,E2,1,E2,2,A=10-10010-1-10100-101,AA%10-10010-100000000!R(A)=2,R()=22:nV,3=2()::V!1,!2,,!n,(!1,!2,,!n)=(!1,!2,,!n)A(!1,!2,,!n)=(!1,!2,,!n)E,En!3=2,!A3=2E(1)|A3|=2n&0,!A,,,,,,∋([1],.[M],1999.[2].[M],1988,3.[3].[M],2002,10.[4].()[M],2001,9.TheEquivalenceoftheRankofLinearTransformationAndCorrespondingMatrixDINGWan-long1,SUNMei-xiang2(1.BasicTeachingDepartmentBaotouLightIndustryVocationalTechnicalCollege,Baotou0104035;2.SubsidiaryMiddleSchoolInnerMongolianScienceTechnologyUniversity,Baotou014030)Abstract:Thelineartransformationisanabstractconcept,obtainingtherankofalineartransformationisdifficult,buteverylineartransformationiscorrespondtoamatrixanditiseasytogettherankofthematrix.Inthearticle,there∀lationanditsapplicationoftherankoflineartransformationandcorrespondingmatrixispresentedtomakeabstractprob∀lemsconcrete.Keywords:lineartransformation;correspondingmatrixoflineartransformation;rank24