線性代數(shù)知識點歸納

第一部分行列式aaaaana2nD((aj)jjn1jjn2aAaAaA1j1i2j21③b***110b***Abbbnn220112200bnnAOAAO=ABOBOBBOAAmn④若B=()ABBOBOa1nOa1naa()aa1n2na⑤n21n21n(n1)2n1aOaOn1n1111xxxn1222x2xx⑥xx21nijjinxxxn1n11n12nabbbabbbb⑦abbba[a(n1)b](ab)n1bbba⑧.⑨對DD與D或D,Dnn1n1n2nnD,D,DDnn1n2n⑩n(1)為nAEASnkSknkkkk10A2第頁共20頁；AA()0；rAn0.M(AA(Mijijijijijij第二部分矩陣aaa1112na21a22a由mn行nAmnm2na1am2aaA或Amnijn()A或AAa.AijA(a),Bb)CAB(c)，ijsijsnijmnb1jbc(a,a,,a)ababab2jij1i2isi11ji22jissjbsjABBA.AB0A或B=03第頁共20頁ABABAn,BABAA1111,nA22BABAn22○○左行00bbbababababa11112n11111211n0b0ab22b2nab221ab222B22122n00abmbm2bmnabm1abmm2abmmn○1○右列bba000abababb1112n1111212mnbb22b0aab121ab222abmB212n22nbbm2bmn00amabab2m2abmmn111A(A)()，④AAmnmnmnmn⑤AAT.AAAAA.TAA.TABTACTTCDBDTTAAA11211A12A22AnATA⑥A*2，為AijijAAnAnn2nAAAE,,AA1.**A1A*n1A*A*(AB*B(BB*4第頁共20頁(A)(A)TATTTTTTT1A111111kk1AA()()n2An111knAAAEAkk0r()n1AA0.ab11db主換位副變號○注：A1cdcaAadbc(AE)(EA)②初等行變換1A1A1A1B1③：BB1BA1AC1AO1OA1A1A1B1OBOBCB1B1a111a11311a2,a④12122a3a31311⑤）ABBAEAB100稱為E(i,j)1Ei,j)1()ijiji(kk()E(i(k))11riik()[,()][,()]EijkEijk(,())Eijkrirkccki,j(k1jij5第頁共20頁○行○左對A對A乘A；乘A.○列○右.A()rAr，，1)；Arr()rAr，的；Ar()rAr，；ArOr()1AOr()00r(A)min(m,n)①A≥;;≤≤mn(A)r(A)r(AA)②rTT③r()r()k0()r(B)nr若A,B,若r(AB)0④的列向量全部是Ax的解ns(AB)(()⑤r≤rArB⑥若、()()()()；PQrArPArAQrPAQ只有零解Axr(AB)r(B)(A)n⑦若r；ABOBOn在矩陣乘法中有左消去律ACBCAB(AB)r(B)r(B)n若r在矩陣乘法中有右消去律.sEOEOr()r與唯一的等價，稱的⑧rrOOOO≤(AB)r()r(B)r(r(B)r(,B)r()r(B)⑨r≤,≤AOOAACr()r(B)rr()r(B)⑩r,OBBOOB6第頁共20頁0B或(II)B6AAE(II)的解法：構(gòu)造初等列變換(I)(AB)(EX)初等行變換BX(II)的解法：將等式兩邊轉(zhuǎn)置化為AXB，TTT用(I)的方法求出X，再轉(zhuǎn)置得XT第三部分線性方程組,,,,,,,2k，kk1knkk12n2112nn,,,,,,是12n12n:,,,12n由mnnaxaxaxb121nn1axaxaxb122n2naxaxaxbm11m22nnaaaaxbn11aaxbAx2n22bmaaaxm1m2mxb11abxaa2212nxbnn7第頁共20頁1122nn()(,)rArAmn12n12s121s則AB2122cc1212nsbnsiiiiA12s12s12sc,c,,c12s12nCABCBAaa1111111122n21ca22212222n22acaamnnm1122m8法29??③④⑤1i)≤≤n12ni12n12n12n12n12nAB.和,,,12n12n12n12n②sn1212n12s1212n④r1212n12s12nr()mA，⑧設(shè)A是mxx112nna12n11aaxbA222jjbm(1),是,也是它的解Ax1212(2)是Ax,對任意k,也是它的解齊次方程組(3),,,是Ax,對任意k個常數(shù)12k,,,,也是它的解12k1122kk(4)是Ax,是其導出組Ax,是Ax的解(5),是Ax的兩個解,是其導出組Ax的解1212(6)是Ax,則也是它的解是其導出組Ax的解2112(7),,,是Ax,則12k也是Ax的解11122kk12k是的解0Ax1122kk12k,,,是Ax12s,,,①12s,,,②Ax12snr()③s.=)(Ab；r(Ab)r()rnnrb；00,12n-rbxkkknrnr01122k,kk.12nr第頁共20頁,,,是Ax,,,√若是1s1sA√Ax與BxAB,r()r(B),rB①②③Ax與BxB√A與BlnPmnQBA與Blnmn第四部分方陣的特征值及特征向量n個nT與bT,)na,a,,b,b,,anabababab1212n1122iinni1,)0.,)2na,a,,aaaa④anT2122212ini1,)1.即長度為1⑤：①：,),)0②：,)(,)第頁共20頁,),),)③：1212(,)k,)設(shè)An和nx,Axx，AxAEA0AE0AEA)()AAE④()A()AOntrAtr⑤A，AA的跡.12ni1⑥n00為AAx0⑦若Aa1a、⑧r()1A=b,b,,bA(ababab)AAA2212n1122nnantrAababab0.,11122nn23n○a,a,,aT為Ab,b,,b為A注12n12n,,,()，fA⑨若A12n①若Af()OA()0fi)()().f②f()fff；fAf(),(),,()()(f12n12n⑩與TAAAAE0i.(AE)x0Aii第頁共20頁設(shè)(AE)x0,,,rr(AE).i12nriii11,kk則Aki22nrnrii,k,,kk12nriA與BPAPBB（P）1A與BP1AP（P）AAA是①EAEB,AB○是A1是BP注00②ABB,AB③A④r()r(B)A與B則Af()B與f()①nAA有nP為APAPA1設(shè)ii12PAP1.nnr(EA)kAn②A為kiiiii()nrAAx0為AA注i③若nA有nA第頁共20頁③若Annii④⑤⑥EA1；T②TAAE；T④A，A1TA⑥A,,12311,)1正交化2(,)22111,),)12313(,)(,)3321122122311233第四部分二次型aaax1112n1a21a22axnn(x,x,,x)axx(x,x,,x)xAxf2n2T12nijij12ni1j1aa1axnn2nn(x,x,,x)AxT12nB,,）A與BCT.（prp2pr(r④⑤與BA⑥r(nóng)()r(B)正交變換n(x,x,,x)x合同變換xfdy2標準形.fT12nii可逆線性變換1第頁共20頁xxii0ij(iijjijkk12n12nTA；；T0fn0A與ECCE；TAPAA；iiT1,BAA