向量组线性相关性的几种判定方法-刘正理

131河南教育学院学报(自然科学版)Vol.13No.120043JournalofHenanInstituteofEducation(NaturalScience)Mar.2004:2004-01-08:(1950—),,,.:1007-0834(2004)01-0015-03刘正理(信阳教育学院数学系,河南信阳464000)　　:探讨了矩阵的初等行变换、向量组的秩、用克莱姆法则或系数矩阵的秩判别齐次线性方程组有无非零解等相关知识在判定向量组线性相关性中的运用,归纳出判定向量组线性相关性的四种方法,研究了四种判定方法之间的关系及应用时应注意的题设条件.:向量组;线性相关;线性无关;判定方法:O172　　　　:A　　、、、.,,.,,,、、.,.:1　α1,α2,……,αm,,k1α1+k2α2+……+kmαm=0,k1,k2,……,km,α1,α2,……,αm;k1=k2=……=km=0,α1,α2,……,αm.2　　　α1,α2,……,αm,α1,α2,……,αmA.r(A)m,;r(A)=m,.3　　　α1,α2,……,αm,α1,α2,……,αmx1α1+x2α2+……+xmαm=0,;,.4　　　α1,α2,……,αm,m=n,α1,α2,……,αmA,detA=0,α1,α2,……,αm;detA≠0,α1,α2,……,αm.1　α1,α2,α3,β1=α1+2α2,β2=α2+3α3,β3=α1+2α2+4α3,β1,β2,β3.　k1,k2,k3,k1β1+k2β2+k3β3=0,k1(α1+2α2)+k2(α2+3α3)+k3(α1+2α2+4α3)=0,,(k1+k3)α1+(2k1+k2+2k3)α2+(3k2+4k3)α3=0α1,α2,α3,,k1+k3=02k1+k2+2k3=03k2+4k3=0,k1=k2=k3=0,β1,β2,β3.2　α1=(2,1,0,5),α2=(7,·15·-5,4,-1),α3=(3,-7,4,-11).　α1,α2,α3A,Ar(A),A=α1α2α3=210　57-54-13-74-11(-1)×r3+r1(-2)×r3+r2-18-41619-4213-74-11r1+r23×r1+r3-18-416017-837017-837(-1)×r2+r3-18-416017-8370000,r(A)=23=m,α1,α2,α3.　α1,α2,α3x1α1+x2α2+x3α3=0.2x1+7x2+3x3=0x1-5x2-7x3=04x2+4x3=05x1-x2-11x3=0A,A=27　31-5-704　45-1-11r1r21-5-727　304　45-1-11(-2)×r1+r2(-5)×r1+r41-5-70171704402424117×r214×r3124×r41-5-7011011011(-1)×r2+r3(-1)×r2+r41-5-7011000000,r(A)=23=m,,α1,α2,α3.3　α1=(1,2,1,3),α2=(4,-1,-5,6),α3=(1,-3,-4,-7),α4=(2,1,-1,0)　A=[α1T　α2T　α3T　α4T],A,detA=14122-1-311-5-4-136-70(-2)×r1+r2(-1)×r1+r314120-9-5-30-9-5-336-70=0α1,α2,α3,α4.　α1,α2,α3,α4A,Ar(A),A=α1α2α3α4=12134-1-561-3-4-721-10(-4)×r1+r2(-1)×r1+r3(-2)×r1+r4121　30-9-9-60-5-5-100-3-3-6-13×r2-15×r3-13×r41213033201120112r2r31213011203320112(-3)×r2+r3(-1)×r2+r412130112000-40000,r(A)=34=m,α1,α2,α3,α4.　α1,α2,α3,α4x1α1+x2α2+x3α3+x4α4=0x1+4x2+x3+2x4=02x1-x2-3x3+x4=0x1-5x2-4x3-x4=03x1+6x2-7x3=0A,·16·A=14122-1-311-5-4-136-70(-2)×r1+r2(-1)×r1+r3(-3)×r1+r4　14　120-9-5-30-9-5-30-6-10-6-12×r4r2r4　141203530-9-5-30-9-5-3(-1)×r3+r43×r2+r3　14120353001060000,r(A)=34=m,,α1,α2,α3,α4.,1.α1,α2,α3,,,k1=k2=k3=0,,β1,β2,β3;2,α1,α2,α3A,A,,,α1,α2,α3.2,x1α1+x2α2+xmαm=0,,,.3,α1,α2,α3,α4A,,detA=0,α1,α2,α3,α4.3.3.,:(1),,,,,1.(2),,,,,2.(3),,,,3.:,;;.,,.[1]　.[M].:,2001.[2]　.:[M].:,1999.SeveralKindsoftheMethodsforJudgingtheRelatedLinearityofVectorsGroupLIUZheng-li(DepartmentofMathematics,XinyangEducationCollege,Xinyang464000,China)Abstract:Vectorsgroup'srelatedlinearityandirrelevantlinerityaretwobasicconceptsoflinearityalgebra.Judgingthevectorsgroup'srelatedlinearityistheemphasesintheteaching,andisalsodifficultpoint.Thistextinducesoutfourkindsofmethodsofjudgingtherelatedlinearity.Studyingthenexusbetweenfourkindsofjudgingmethods,andthequal-ificationwhichmustbeattentioninapplication.Keywords:vectorsgroup;relatedlinearity;irrelevantlinearity;judgingmethod·17·