线性代数期末考试重点

精品文档1n精品文档放心下载n((2A(ijM312)1）

AB=AC）

AB=0或B=0谢谢阅读A与B(AB)TATBT，(kA)TkAT,(AB)TBTAT谢谢阅读设(x)a(A)aEa01Aa0naxxA为nan1nAn称为AnAPP(A)aEa101AanAnPa1PaP1PanP1=P()P101n谢谢阅读diag(a,a12,an)()diag((a),(a12),(an))nA,BABEA,B感谢阅读nAA0；r(nR(nA谢谢阅读A与E等价；AAA1欢迎下载。精品文档A11AA*AE1初等列变换A,EE,A或谢谢阅读EA1，EAAA是n若AA1(A1)1A谢谢阅读若AAT(A)T1(A1)T1k若Ak0kA(kA)A11若BB(AB)1B1A1AA,B为n精品文档放心下载①ATA②AA③ABn感谢阅读AAAAA*1121AA1222n1An2AAAnn1n2nAAAAAAE**①A*An1②A*AA1③(A*)11AA④(A*)1(A1)*等E感谢阅读E(i,j)Ei、j感谢阅读E(i(k))（E的第ik）感谢阅读E(ij(k))（把E的jki精品文档放心下载感谢阅读E(i,j)1E(i,j)、E[i(k)]1E[i(k1)]、E[ij(k)]1E[i,j(k)]；谢谢阅读。2欢迎下载。精品文档:感谢阅读rirji与j行k0rikk第ikrikrj第j行ki.感谢阅读Amn谢谢阅读①对AAm②对AAnAABAA与BA与B谢谢阅读感谢阅读谢谢阅读感谢阅读设B为mn①A与BmPPAB谢谢阅读②A与BnQAQB精品文档放心下载③BmP，nQB精品文档放心下载AXB，XA1B(AB)(EAB)谢谢阅读初等行变换1XAB，XBA1(ATBT)(EX)。感谢阅读初等行变换Tr(A)或（）感谢阅读A，R（谢谢阅读初等行变换若A是mn0R(谢谢阅读②R(A)R(A)③A~BR(A)=R(B)精品文档放心下载TA为mnPm,Qnr(r(PAQ)max{R(A),R(B)}R(A,B)R(R(B)感谢阅读若BmnR(AB)R(R(B)精品文档放心下载⑦R(AB)min(R(A),R(B))若ABmnntOR(R(B)n3欢迎下载。精品文档为nA谢谢阅读A1A2①AAiA2As若AiAsA11A12(i,s)A0A:11A设A是m,B是n且Aa,Bb则OABO1mnabsACHBm阶、nHOB感谢阅读H1A1A11OB1AOHBmn精品文档放心下载CBHH1A1BCA11OB1．：,,m,12,k,,kkkm11kkmm12

22：,,,,,,mm

1212=1122mmAA精品文档放心下载Axxxm解m1122（,,,）的秩等于矩阵B=(,,,m1212有m，的秩4欢迎下载。精品文档：,12,,m及B：,12,,s，若BABAAB精品文档放心下载（,12,,m（,12,,s）ABAB谢谢阅读A与BAB感谢阅读：,b,b,a,aB=AKbA:a谢谢阅读1212

lmAX=BR(R(A,B)感谢阅读A:a,a12,amBb1,b2,blR(R(B)R(B)，,b,bb12,b,bb12l,aA:a12)a,a,am)12l,am：,k,,k,,,mkm12120Ak11kkmm

22,k,,k,,kmm,12

121）向量组,12,,mA=（,12,,mm；R(m个n1a,aaaa1112aaa2n,,a1n，2(a,a,,a)2n,a)n(a,an20，行21220aa

an2nnm个nnmn1个n精品文档放心下载：,,,B:,,,,mm1212m15欢迎下载。精品文档BA感谢阅读：,12,,B:,,,mm12,A6）若r,12,,mnr到的n维向量组,,,11112

mA：,12,,mm1性谢谢阅读A：,,B:,,,sts,12向t12：,,s,12:：,,,B,,,ss12t）t12Ar个向量,,,r12，A0：,12,,rAr1Ar1谢谢阅读,12,,rAA,:12,,r精品文档放心下载谢谢阅读④若向量组,12,,s⑥向量组,,s,12,12,,srAr（,12,,s）,12,,s,12,,,,,,,,s1212tt6欢迎下载。精品文档r(,12,,s)r，r(,12,,t)prp谢谢阅读V为nVVV感谢阅读1,nVx,R谢谢阅读,（2---设Vr,12,,r①V,12,,②V,,,rr12则称向量组,12,,rVrVV为r维,a,a,b,bARab3（a,a331212B,bP=A1B（b,b3阵1212,a3xy11y1x2y2xn，nx，yxyx,yx221xynnx,y称为向量x与y.当x与yx,yx谢谢阅读Ty.x,y,z为n感谢阅读①x,yy,x；②x,yx,y；谢谢阅读③xy,zx,zy,z.当xox,xoxox,xo谢谢阅读⑤x,y2x,x[y,y]感谢阅读x=x,xxx，xnx.x22212

n当x,y0x与y.谢谢阅读.精品文档放心下载若n维向量,12,,,12.rr7欢迎下载。精品文档,aa12,ar取b1ab,b12b2b,ab.r1rbr1,br1r1．aaabxxx1111221n1naxaxaxb2112222nn2Aaijxmnaabaxx12mnnmaa1112aaaaaab1n11121n1aa2122aabaa2n，A=21222n2m2aamnaabAxbm2mnm1aaab1baaa22n，,，b2nam2amabxbxx11nn22nAmnxbr(r(r(r(nr(r(A)n谢谢阅读aaa0

xxx111122

1nnaxaxax02112222nnaxxxaa012mnnn元齐次线性方程组仅有零解r(n谢谢阅读nr(n感谢阅读An元齐次线性方程组仅有