吉林大学线性代数线性代数192课xm44

1、第四章 第四节线性方程组的解的结构回忆之前的定理( )( )( )( )( )ar anar ar bnar ar brnx = 0 x = bx = b有非零解有唯一解有无限多个解11 1122121 122221 12211121121222212111221111211000,nnnnmmmnnnnmmmnnnnna xa xa xa xa xa xa xaxaxaaaxaaaxaaaaxaxxx1xx = 0 x22221,()2,()()aaaaaakaa kk ak11111111xxx = 0 xx = 00+0 = 0 xx = 0 xx = 00 = 0性质 ：都是的解，则也

2、是的解证明：性质 ：是的解也是的解证明：021220ss,sttkkk11t x是解集，：是最大无关组。全体解都可以表示成称为方程的基础解系1112121202000000010 , 23 , 43010252xxcccccc 111,1,11111,11,111121112210010000100n rrr n rrn rnrrrr n rnrrrrnabbbbabxb xbxxb xbxxbbxbxccxx x = 01,2,122001001n rrr n rn rn rn rbbbcccc1x1222222,(,)|0(,),n rn rn rn rn rn rn rcccnrernr

3、11111x 全体解都是的线性组合阶行列式是方程的基础解系。1212111121,12,111,100010,001,100rrnrrnn rrrrr n rrxxxxxxxbbbxbbbbb 1代入一组线性无关的向量121,2,2,001001n rrr n rn rbbbb自由变量可以任意赋值，不同赋值得到不同的基础解系123456,789(32)1121sxynzr arxyczr01234123412341342340253207730111111112532 07547731014108231077111154 0754 01770000000023775477xxxxxxxxxxx

4、xaxxxxxx例题12341221212343410,01232377775454, ,( ,)77771010010111,11xxxxccc crxxxx 1121212345151-77779191, ,(,)777711111-11-1xxkkk krxx12312124121212123411111111253243107731862052014310000043,(,)521001,( ,)43521045axxxx xxxxxxcccxrcx 任意01,32 不是行最简也可以求解秩的性质()()(),(1,2()( )( )( )( ),)m nn lisssabobaaila

5、srrr brr arnr ar bns12l12li12lb ,bbb ,bb0,0,0b = 0 x = 0bb ,bb解集ba( )( )r br a能由 线性表示( )(4:(1)( )ssr ar br anrr bnrabx = 0 x = 0例题，同解，求证15:()( )()()()()()0() ()0()( )tttttttttr a ar aaa aaaaa aa aa aaaar a ar ax = 0 x = 0 xx = 0 x = 0 x = 0 xx = 0 xx =xxx = 0例题证明只需证明与同解如果 满足如果 满足由于两个方程组同解，得到非齐次线性方程组

6、解集的性质121212123: ,()4: ()aaaaaaaaaaax = x = x = bx = x = 0b-b = 0 x = x = bx = x = 0 x = x = b0+b = b性质都是的解， 则是的解。证明：性质是的解，是的解， 则是的解证明：22*22,12304560 , 789012314564 , 7891217akakxxyyczzxxyyzz 111111x = 00 x = b 100121c 11*1,(,)n rn rn rn rn rn raaakkakkkk111x = 0 x = bx = x = 0 xx = bx的解特解任意解通解：通解：任意实数， ，齐次方程组的基础解系1234123412341243403112321110121111011110111131 00241 0012211000001123001222( )( )212122xxxxxxxxxxxxbr ar bxx