线性代数第五习题答案详解

第五章n维向量空间习题一．==TT=T==TT=TT．=123=123=123=12333=123116]=6123115a+a=a213263115将aa=a+aTTT123213263.T．V…,0)V知V设)V)V,11112n112n1x+y+12n12n)+y+1122nn12n12nx)Vk1122nn112n12n)VV.12n11Vx)Vy)V但x,22n22n222x)VV.nn22习题二．ba1234ba1234aaaa11223344,k,TTT123412abaTT341234TTTTT1234kkkk01234kkk21223kk1k101k==1234ba1234a.12342k3kkk12342kk2k1k12334kk1212kkk12kk213k4ba1234.12342知a1234.111111111126015015aaa123133022004Raaa3123a.123111111111241406012aaa123317024000Raaa23123a.123111111111103014014aaa123112023005Raaa3123a.123k使123kbbb112233b112123123ka112123123即+a123123233a123kkk0k012310k0kk232k0k033b.123k使1234kbbbb11223344b=a=a112223334441k112223334441+，141212233344法a1234a1234k，kkk14212334kkkkk214131取k00k使11234kbbbb11223344b1234akkkk在一123414212334kb12341234a12340kk14k0k12，kk023kk03410011100001100011b1234b0k,12m12mkbbb1122mma1a2a)mb…a1b…a2b…a)m由kbbb1122mmkaa+a12mkaa+a12mkaa+a12mkaa+a12mkaa+a12mrk12m即kaaa1122mmkaa12m12m12m.习题三．012301231340135001201350000.100210010101021002011110111000101000111100120021.a：1231411411412130950951540950003670000a12100a1201230030A.a1234511221021512031311041110221211122102151000000022251A021510022211221021510022200000AA:a,a,,aB:b,b,,b；()()RARBr设A12s12tC:a,a,,a,b,b,,b12s12tCAACRC)R()rR(B),b,,bB的极大无关组为：bC1i2ir,b,,bCbCB1i2irABABB和Ar,12r12rA与Bab12r12r.BAbaa+a112rbaa+a212………………rb=caa+arr11r22rrrk使12rkbbb1122rrccccc+b+ccb12rr1112rr2212rrrr由a12rkcc+c12rr1kcc+b12rr2………………kccb12rrr把kk0b要k12r12r12r12r00即cijc11ccc12rcc21222rcccr1r2rrcccc1112rc22c2rCC设C=21-1-1cr1cr2crrba11ab22则barrb1a1ba-12=CC2-1barr即a1b1ba2=C2-1abrra=cb+cb++cb112rra=cb+cb++cb212r2r………………a=cb+cb++cbr12rr1r2ab.12r12r.若an个na12naaa.12n12n.nane,12n12a,a与en12n12n12na.12nRe3123123123e与aae123123123123123ea.123123011011a＝,a,ae,e,e101101且312312110110112212011111101212211011222112212111即e,e,ea,a,a＝212212312311222ea)=R.31231231231100110010111011aT11011011101110111aT2213320221011bT1011101110111bT2abTT1与1abTT22,ab,baTTTT1212ab1212.1212kk)1kk3即21212420111110kk1kk=3430123101kk111kk3=3124.1212.121211)121231113101111223131221122)31121222.1212)1212k,k使123kaaa112233即k2kk0123kkk0123k2k023k123a即a为R.3123123baaa,baaaba1112233241526312345612123lll5lll9123456lll0lll8和123456ll7ll2356l2l3143l3l和25l1l236ba和12123VV中ee,23e=e有n23n23neeee为V.2233nn23n习题四112121111211211024013101030103221200340034003443xx4x13x2344xx34取x4433=431,k1211121120136130040001051015004000002xx1x3x420x10,取201x42110,==102001,k1122121247231512470710143127074300164136097112470790051247071014001053270070.231412451021124507701123820280000419607700002x1x13xx2332313==011=0211211211121301211121103112121120330101101100000xx1133xx221=0111==010.a和a由,12a2aa00121314a2aa111213a