考研数学之线性代数讲义(考点知识点+概念定理总结)

收集自网络，不以任何盈利为目的。欢迎考研的同学，下载学习。线性代数讲义06,07年考题第一讲基本概念axxx=b,12n1axxx=b,212…………naxxx=b,12nmnmn,xki12nib12mn由mnm行nmn2011111022549333845aa…aaa…ab1=aa…aaa…ab2…………aa…a………aa…abm为其系数矩阵和增广矩阵.ij0A和B由nna,12na1,a,12n2┆annn1mnnmA12n12n0mnA和BmnBmnAcmncA①.②③.④.⑤或A=0.mnAnmAA(或T①A)=.TT②)=AB.TTT③)A.TT当TTnc12s12sc1122ssc12s1nn2snnn把nn0的n1cc.0的n0的nAAnTAA0的n阶T①②0③①②00请注意:①②0③B(00.BA是nAAAAAAAAAAAAABAA和BB第二讲行列式nn行nn2aa…aaa…a……….aa…a,…,…12n1n2n2nAnaa=aaa.naa…a………aa…aaaa,1j2jnjj12n12(jjjj…j则12n12n(()1212n((jja.=2j21jjjn………aa…ann12n0用它作行列式的计算.例如对角行列式,上(下)三角行列式的值就等于主对角线上的元素的乘积,因为其它项把nijaM.称1①A|.T②|n③1212④A*=AOOB*B1a11…1a2aa…a13na1a…a222232n……………aaaa123na312n(aa).jiij0a,a123nnnnD12nDDii0AA例1①2aaaaa2aaa②111111③111222aa2aa.aaa2aaaaa2111.11333.4441例2123452345134512.4512351234例31111111.211131114例4a0bc0acb.bca0cb0a例5a000a000a0.00a000例6x135132936例7求a450的x和x3402b121x例8设4阶矩阵,,,,|=|.1abcd23123例9xA1y0例1030402222000532n例11aaa…aa123bc0…00n12n(bLbacLc.0bc200=i1311ii1ini1…………000…bc1n-1n例12aaa…aa012bc0…00n11nncLcabcLc.b0c…00=ac22011iii1iini1i1…………b…0n1nn例13b0…00ab…00abn1n1n…………=abiniababi0000…b000aj11例14x123,222123123例1x③a433例2例3xxxx+xxx+xxx+xxx+xxx.1234234134例4124123例5.4235例6例7例8例9例10例14x123第三讲矩阵ABA和B.ABABABAiBj设aa…abb…bcc…c=cc…c=aa…a=bb…b………aa…a,bb…b,cc…c,则cbbb.①②③由=0或由AC和0由CA和0.①AB,.②③=④)=BA.TTTnnA和BABnA和B设knA的k次方幂A即k个AAE.kA①AA=A.0kh②A)=A.kh)和ABkkkn设xxnAmm10AAaA.mm10A.nn当A和B)=AB;222AB22B)CAB(A1A和BnnA和ABAABB=ABABABABAABBABABABABABA0…01=0A…02………00…AnAAA12kA0…0B0…011=0A…0,=0B…02………2………00…AA和B00…BkkiiAB0…011AB=0AB…0.22………00…ABkk设A是mn矩阵B是nsAB,AB12n12s12s①ABii即12s12s②)=b.T12n1122nn).ABi是AB的第iTiiI12ni12niABiBAikEkCA,.131=20.12EE的0数c乘EiEiEjciicj也EXA..A0的n当B只有一列时,(I)就是