????????????????????????1.????????????????????????;2.??????????????????????;3.??????????????????????.????:1.????n??????????????a11x1+···+a1nxn=b1?·a··············m1x1+···+amnxn=bm2.??????????????????????????a??aa?11a12···a1nb12122···a2nba··············2??·?m1am2···amnbn??.??????????????:1.??m=n????????????????,????(????)??????????????????,??????????a;11,···,ann??????????,??????????????2.??????????E????1,??????????????0??n????????????????,n;3.??????????????????????????,????O,??????????????m??n????,??????Om×n;4.????:????????A??B,??????????????????,????????????,A??=??(a????????????,??ij)m×n,B=(bij)m×n,aij=bij(i=1,2,...,m;j=1,2,...,n),????A??B????,????A=B.5.????????.????????????????????????,?????????????????1.????????.?a11?????0a22..?.0??.ann2.?????,??a11=a22=···=ak??nn=k??,????1?????????,?????0k...0???=k???.k01..0???=kE.13.???????????,??a11?k???????ak11?0a22..?.0??=???ann0ak22...0????.aknn??????????????.??????????:Example??1.??????????????????32xx++33yy==212.??Gauss??????????????:x=1,y=?13.3.??????????????????????????????????????,??????????4.32332101??????????1r?1????→r2231r?2????2??×?→r11003?111?3?×?→r21001?1??13.??x=1,y=?13.????:1.???????????????????????????????????2.????1.??m×n????aij(i=1,2,...,m;j=1,2,...,n)???????a??????am·??(?????)n??(????)????????a1121a12···aa1nn????????,????a···m1a·22······2·n·??,????????m??m2·····amn??(aij)m×n.3.??:??????????????A,B,C,...????????.m??n????????A??????Am×n,??????????????????????????????,??aij??????????i????j????????.??????????????????????????????.????????:?1?1.????????.????01..0??.??12.??????????Am×n??An×m,??Am×nEn=An×m,EnAn×m=An×m.3.????E????n??”????1”.????????????,??????????????????????4.????????????????,??????????????????E.????????.????:1.????????????????????????????2.?a11a???a12···a1n22···a2n??1a0.....?,??112·?····a1na.?0..·a2n?....??.nn13.??????????????????????????????4.?a??a1101021.a22??....?,??a21.1...??.a...·.??.·a....n1an2·nnan1an2···1??.??????????????:1.??????????1.??????2.????????????A=(aij)m×n,B=(bij)m×n,????C=(cij)m×n=(aij+bij)m×n????????A??B????,????C=A+B.2.????.??????????????????????????????.3.??????????????????????????:????????:A+B=B+A;????????:(A+B)+C=A+(B+C);??A+O=A,????A,B,C,O????m×n????.3.??????????????:1.????3.??λ??????,????A=(aij)m×n,????????(λaij)m×n????λ???????A??????,????λA,??·λA=(λaij)m×=?λa11λa12n?λa··21·λa··22····λa?λa······λa1n2n?m1λam2·λa···?.·mn2.????????λ,μ????????)????????????:(A,B????m×n????,????????:(λμ)A=λ(μA)=μ(λA);????????:λ(A+B)=λA+λB;??1·A=A,(?1)·A=?A.3.????????????????????????????????????.??.????????????????:1.????4.??????A=(aij)m×s,B=(bij)s×n,??????A??B??????Cm×n???...?C=AB=(c=?·a···········??.?..b1j?....ij)m×n.?··i1·a··i2·······a··is??.b2j·??.....?......??..bsj..?.=(ai1b1j+ai2b2j+···+aisbsj)m×n.2.????????:,????????????????????????????????.??????????????????????????????1.??3.??????A=120130,B=?42111??AB??BA0,??????.????2.??.AB=12013??0?42×321110??3×2??=12××44++01××((??1)1)++30××2212××11++01××11++30××00??=10??713,2×23.??????????BA=?41??21110=611123×2201302×3210?63.3×34.??????3????????????????????????,????????A??B????????,??B??A????????.2.??????????:1.??????????????A=(aij)m×n,????????(?aij)m×n??????A????????,?????A?,??·?A=(?a??ij)m×n=???aa11?a1221?a22···??·a··········?··?a·?·a1na·2·n???.m1?am2mn2.??????????,????????????????A?B=A+(?B)=(aij?bij)m×n.3.????:??????????????????????????????.????1:Example??1.??1.??A=43?1??????301,B=15??11?03,??3A?2B.2.??.??3A?2B=34????????330?11?2112??5??1??1?03=99??0?33?????102??22?06=?101112?93.Example1.????2.??????????????????????????????????????????????A=aa11a12a????a21a22a13a14????131a32a23a2433a34????232.?????????????????????????????????????????b????11b??B=??bb21b12?????b31b22?32?,????141b????2????3414????3.????C=AB????C=AB=cc11c12????c21c22????131c32????23??????????????????????,??????????????????????.Example1.??4.????????A=?11?1????1,B=?22?1????1,C=21?3??3,D=??12?5??5.????:(1)AB=O;(2)AC=AD.2.????.AB=???????11?11?2??2?11=??0??000=O.3.AC=???????11?112??=??1?33?3??300,AD=???????????11?1112?55=?330??0.??AC=AD.4.??????4????????????????????????,??????????????????????????????.????????????:1.??A??n??????,????????????????,??k??A·A=A2,A·A·A=A3,??A·????A...A??=Ak.2.??????(????k,l??????????)Ak·Al=Ak+l,(Ak)l=Akl.3.????????A,B????n??????,??????(AB)k=AkBk.k??4.????????(AB)k=??AB·AB????···AB??=AkBk.1.????5.??.??A,B??????????????,C=AB,????C????????2.??a.??A=(0a.ij??)n,B=(bij)n.??????,??i>j??,ij=bij=c??ij=(0,···,0,aii,···,ain)(b1j,···,bjj,0,···,0)T=a0ii,bii,ii>=j.j????????.3.????,??????????(??????????):1????????????????????????????????????;2????????(??)????????????;????(??)????????????????????3????????????????????????????????.4??????????????????????????????.??.?????????????????????:??a11x1+a12x2+···+a1nxn=b11.???????????a21x1+a22x2+···+a2nxn=b2?·a········.a···m1x1+m2x2+···+amnxn=bn?·2.??A=?a?a11a12·21··a·22·······a???a1n?b1x1?2n???b.2??,x=??x.2??a·m1am2····?,b=·a···?.b.??..mn.?mxnA????????????,b????????????,x??????????????.3.??????????????????????????Ax=b.??????(A,b)????????????????????,??????????????????????????.4.??b=0,??????????????????????????,????????????????????????.??????????????????:1.??7.??????????????2x+x2?5x3+?x11?3x2?5x4x4=8,4=9,??x1+2x42x?2?x37x+23+x46x=5,4=02.????????????????2??1?13?50?458?9?0?.124??17265?0??????????????????????????:1.??????:????????????????:ab=ba;??????????????:AB=BA;2.??????:????????????????:ab=ac(a=0)?b=c;??????????????:AB=AC??B=C;3.????????????????:ab=0?a=0??b=0,??????????????:AB=O??A=O??B=O.4.??????:??????????????????????????,??????????????????????:(AB)C=A(BC);??????:A(B+C)=AB+AC;λ(AB)=(λA)B=A(λB),λ??????.??1.????6.??aT=(1,2,3),b=?14??,??aTb,baT??(baT)202.??2.??.aTb=(1,2,3)?14??2=8.????????3.baT=?14(1,2,3)?418122=2?24?36.4.b(aT)20=(baT)(baT)···(baT)=b(aTb)(aTb)···(aTb)aT=819baT.??????????????????:1.?????????????????,m??????n???????????????????????a?a11xx1+···+a1nxn=b1211+??·a·············+··a··2nxn=b2m1x1+···+amnxn=bm2.??m?×(n+1)??????(????????)??a??a11a12···a1n||b121a22··???????????????.a··········a·2·n·||||·b·2·m1am2···amn||bm3.????????????????????????????????????.??.??????????:1.????5.??m×n????A??????????????????????,??????n×m????????A??????????,????AT??A??.2.?????????2?A=12345678=?AT=?1?3542×476??.84×23.??????????????????????????(AT)T=A;??(A+B)T=AT+BT;??(λA)T=λAT,λ??????????(AB)T=BTAT.
【线性代数 大连理工】第一章
