425130011 0010 1010 1101 0001 0100 10111一、非齐次线性方程组解的性质二、线性方程组有解的判定条件三、线性方程组的解法四、小结 Page 2 AX=0.\ue007齐次线性方程组称为非齐次线性方程组AX= 的导出组(或对应的齐次线定1性方程组\ue002义下面讨论非齐次线性方程组与其导出组的解的关系. Page 3(2)如果u1,u2是Ax=b的两个解,则u1-u2是Ax=0的解.证:∵Au1=b, Au2=b故A(u1-u2) =Au1-Au2=b- b =0(1)如果u1是Ax=b的一个解,v1是Ax=0的一个解,则u1+v1也是Ax=b的解.证: ∵Au1=b, Av1=0故A(u1+v1)=Au1+Av1=b+0 =b Page 4由关系(2)知 v1是Ax=0的解而 u2=u1+v1即Ax=b的任一解是u1与Ax=b的某一个解之和.取 v1=u2-u1证:由关系(1)知u是Ax=b的解.反之,对Ax=b的任一解u2,要证明u2一定可以写成u1与Ax=0的某个解之和.定理1 若v是齐次线性方程组Ax=0的通解,则u=u1+v是Ax=b的通解.是非齐次线性方程组Ax=b的一个解,1u Page 5:s n,RnAX\ue007\ue00b\ue005 元非齐次线性方程组定理21)) 23) ( )\ue007\ue002其中( );AR AR A b)\ue004R无解的充要条件\ue000\ue001;,(nbA\ue005\ue005有唯一解的充要条件,R AR A brn\ue005\ue005\ue004有无穷多解的充要条件是\ue003112212,的一个,,,为AX=0AX=.n rn rn rXc XXc XXcXXX\ue007\ue003\ue003\ue003\ue002\ue002\ue00200AX= 的通解为 为导出组基础解系\ue003的一个特解 Page 6B,bbAX的方法\ue006把增广矩阵A=(A, )初等行变换\ue007\ue007\ue005上述定理告诉我们判断非齐次线性方程组是否有解\ue003以及当有无穷解时求解化为Jordan阶梯形10... ... ... ...000... ... ... ...0不妨设为...1121...r1,2,121,101...000......00...0...r n r...rr00000010000...000...00...0000n rn rbbddbbdBd\ue003\ue003\ue003\ue002\ue00f\ue010\ue010\ue010\ue010\ue010\ue010\ue010\ue010\ue010\ue011\ue019\ue01a\ue01a\ue01a\ue01a\ue01a\ue01a\ue01a\ue01a\ue01a\ue01a\ue01b\ue005\ue010\ue010\ue01a Page 7由于初等变换不改变矩阵的秩\ue003故d1当时100( )( )( ),( )rR AR AR A AXR A\ue007\ue002\ue00c\ue002\ue005\ue00c\ue005\ue005当时无解\ue007\ue0071dd00( )( )( )( ),,rR AR AR AR Arrn AXn AX\ue007\ue007\ue002\ue0051\ue005\ue005\ue005\ue005 \ue004\ue005\ue005\ue005当时有唯一解\ue007r\ue002\ue005当时有无穷多解.d,rr AX\ue005\ue005有解\ue007 Page 8\ue018\ue018\ue017\ue018\ue018\ue016\ue015\ue005\ue003\ue002\ue002\ue0021\ue005\ue002\ue002\ue002\ue003\ue005,\ue0035\ue002\ue002\ue002\ue005x\ue002x\ue002x\ue0022x\ue0021x.123143823x7622x, 2323, 75543121432435432x1xxxx1xxxxxxx解\ue01a\ue01a\ue01a\ue01b\ue01a\ue01a\ue019\ue010\ue010\ue010\ue011\ue010\ue010\ue00f\ue003\ue00323\ue003\ue0051213438621202312131B例\ue000求下述方程组的解 Page 9\ue01a\ue01a\ue01a\ue01b\ue01a\ue01a\ue019\ue010\ue010\ue010\ue011\ue010\ue010\ue00f\ue003\ue003\ue003\ue003\ue0030\ue000 \ue00100000R0000002362120711111~.,知方程组有解由BAR所以方程组有无穷多解.且原方程组等价于方程组\ue015\ue002\ue005, 3, 2\ue005\ue003\ue005rnAR又\ue017\ue016\ue003\ue002\ue005\ue00223\ue003x\ue0034\ue003\ue0053\ue00226227554321xxxxxxxx Page 10求基础解系.1x00,0\ue00310,001543\ue01a\ue01a\ue01b5\ue01a\ue019\ue010\ue003\ue010\ue011\ue010\ue00f\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue01a\ue005 \ue003\ue01a\ue01b\ue01a\ue019\ue010x\ue010\ue011\ue010\ue00f\ue005\ue01a\ue01a\ue01b12\ue01a\ue019\ue010\ue010\ue011\ue010\ue00fx\ue015\ue017xx令依次得.32,10,212121\ue01a\ue01b\ue019\ue010\ue011\ue00f\ue003\ue01a\ue01b\ue019\ue010\ue011\ue00f\ue003\ue01a\ue01b\ue019\ue010\ue011\ue00f\ue003\ue003\ue005\ue01a\ue01b\ue019\ue010\ue011\ue00fxx232423456xxx\ue002xxxx\ue002\ue016\ue005\ue002代入导出组 Page 11.10032,01010,0012121321\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue005\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue005\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue003\ue005\ue00a\ue00a\ue00a求特解.223,29, 021543\ue005\ue003\ue005\ue005\ue005\ue005xxxxx得令所以方程组的通解为故得基础解系 Page 12.00022329100320001003012121321\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue0117\ue010\ue010\ue010\ue010\ue00f\ue003\ue002\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue002\ue01a.\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue0111\ue010\ue010\ue010\ue010\ue00f\ue003\ue002\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue0102\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue003\ue005kkkx,,1为任意常数其中kkk另一种解法\ue01a\ue01a\ue01a\ue01b\ue01a\ue01a\ue019\ue010\ue010\ue010\ue011\ue010\ue010\ue00f\ue003\ue00323\ue003\ue0051213438621202312131111B Page 13\ue01a\ue01a2\ue01a\ue01b\ue01a\ue01a\ue019\ue010\ue010\ue010\ue011\ue010\ue010\ue00f\ue003\ue003\ue003\ue003\ue00309000000000002362120711111~\ue01a\ue01a\ue01a\ue01b\ue01a\ue01a\ue019\ue010\ue010\ue010\ue011\ue010\ue010\ue00f\ue00323\ue0030000000000002312110202101~则原方程组等价于方程组 Page 14\ue018\ue015\ue017\ue018\ue016\ue018\ue016\ue015\ue002\ue003x\ue0032\ue003\ue0051\ue00335\ue002\ue003\ue0052232212x\ue003392x\ue0022159x42x32x531xxxxxx\ue018\ue018\ue017\ue018\ue018\ue018\ue005\ue005\ue005\ue002\ue003\ue003\ue003x\ue005\ue003\ue005\ue00d554433543232232xxxxxxx所以方程组的通解为 Page 15.0002232910032010100k012121321\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue002\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue002\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue002\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue003\ue005kkkx.,,321为任意常数其中kk Page 16例 \ue005 求解非齐次线性方程组\ue015\ue002解对增广矩阵B进行初等变换\ue003\ue010\ue011, 2\ue018\ue017\ue018\ue016\ue005\ue003\ue002\ue005\ue003\ue002\ue003\ue0054\ue003\ue002\ue0031x. 3222, 2353, 1324321324321xxxxxxxxxxx\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue00f\ue003\ue003\ue003\ue003\ue003\ue0053(22)1R22351311321B1233122rrrr \ue003, 3rr\ue003\ue003\ue01a\ue01a\ue01b\ue01a\ue01a\ue019\ue01a\ue01b\ue010故方程组无解\ue004\ue010\ue011\ue010\ue010\ue00f\ue010\ue011\ue003\ue003\ue003\ue003\ue0031045010045000113210\ue01a\ue019\ue010\ue00f\ue003\ue003\ue003\ue00320104511321)(\ue005\ue005BRA显然\ue003 Page 17例\ue001求出它的一切解\ue004在有解的情况下\ue003. 0是有解的充要条件证明方程组5432151a5a454343232121\ue005\ue002\ue002\ue002\ue002\ue018\ue018\ue017\ue018\ue018\ue018\ue018\ue016\ue015\ue005\ue003\ue005\ue003\ue005\ue003\ue005\ue003\ue005\ue003aaaaxxaxxaxxaxxaxx解证对增广矩阵B进行初等变换\ue003方程组的增广矩阵为 Page 18\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue00f\ue003\ue003\ue003\ue003\ue003\ue005543211000111000011000011000011aaaaaaB\ue01a\ue01a\ue01b\ue01a\ue01a\ue01a\ue01a\ue01a\ue019\ue010\ue010\ue011\ue010\ue010\ue010\ue010\ue010\ue00f\ue003\ue003\ue003\ue003\ue005514ia3210000011000011000011000011~iaaa\ue000 \ue001\ue000 \ue001051\ue005\ue00d\ue005\ue005iiaBRAR\ue000 Page 19. 0x51x\ue005\ue005\ue006iia是方程组有解的充要条件由于原方程组等价于方程组\ue018\ue018\ue017\ue018\ue018\ue016\ue015\ue005\ue003\ue005\ue003\ue005\ue003\ue005\ue003454343232121axx5axxaxxa由此得通解\ue006\ue015\ue018\ue018\ue017\ue018\ue018\ue016\ue002\ue005\ue002\ue002\ue005\ue002\ue002\ue002\ue005\ue002\ue002\ue002\ue002\ue00554454335432243211xaxxaaxxaaaxxaaaax.5为任意实数x Page 201234234344111110aa\ue002a\ue002aaa\ue002aaaxkka\ue002\ue002\ue002\ue012\ue013\ue013\ue013\ue013\ue013\ue013\ue01c\ue01d\ue01d\ue01d\ue01d\ue01d\ue01d\ue012 \ue01c\ue013 \ue01d\ue013 \ue01d\ue013 \ue01d\ue013 \ue01d\ue013 \ue01d\ue013 \ue01d\ue005\ue002\ue014 \ue01e\ue014\ue01e即为任意实数 Page 21例\ue002 设有线性方程组\ue015\ue018\ue017\ue018\ue016\ue005\ue002有无穷多个解\ue002\ue005\ue009\ue002\ue002\ue005\ue009\ue002\ue002232?B \ue0051有解32\ue0091321\ue009x1\ue009xxx,xxxxx?取何值时问\ue009解对增广矩阵\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue00521111111\ue009\ue009\ue009\ue009\ue009B\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue009\ue00f1111111~2\ue009\ue009\ue009\ue009作初等行变换\ue003),(bA Page 22\ue01a\ue01a\ue01b\ue009\ue01a\ue019\ue010\ue010\ue011\ue010\ue011\ue010\ue011\ue00f\ue010\ue00f\ue003\ue003\ue0031\ue003\ue009\ue003\ue009\ue009\ue003\ue00922221110111011~\ue009\ue009\ue009\ue009\ue009\ue009\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue00f\ue010\ue003\ue003\ue002\ue003\ue003\ue003\ue009\ue003\ue003013222212011\ue00910~\ue00921\ue009\ue009\ue009\ue009\ue009\ue009\ue009\ue000\ue001\ue000\ue001\ue000\ue001 \ue000\ue001\ue000\ue001\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue002\ue003\ue002\ue003\ue0031\ue003\ue0030\ue00521210110\ue009\ue009\ue009\ue009\ue009\ue009\ue009\ue009\ue009 Page 23\ue000 \ue001,1\ue010\ue0111时1当\ue009\ue005\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue00f00000000111~B., 3 方程组有无穷多解223\ue004x\ue005BRAR其通解为13231xxxxxx\ue005 \ue003\ue003\ue015\ue016\ue018\ue018\ue017\ue005\ue005.,32为任意实数xx Page 24\ue000 \ue001,1\ue010\ue0112时当\ue009\ue00c\ue000\ue001\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue00f\ue002\ue002\ue003\ue009\ue00322120011011~\ue009\ue009\ue009\ue009B这时又分两种情形\ue006\ue000 \ue001,2\ue000 \ue001\ue005:, 3,2\ue003) 1方程组有唯一解\ue000时\ue0051\ue002\ue005\ue003\ue005\ue00cBRAR\ue002\ue009\ue001.21,212321\ue002\ue002\ue005\ue002\ue009\ue009\ue009\ue009\ue009xxx Page 25.,故方程组无解BRAR\ue00c,2) 2时\ue0031\ue005\ue009\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue003\ue003\ue00330006330421~B Page 26nnBBRRAARR\ue005\ue004\ue005\ue005\ue00d\ue00d有无穷多解.bAx\ue005非齐次线性方程组bbAx \ue005Ax齐次线性方程组\ue000 \ue001\ue000 \ue00100只有零解0有非零解\ue005\ue005\ue005AxAxAxnnAARR\ue005\ue004\ue00d\ue00d;.;有唯一解bAx \ue005( )( )R AR B\ue00c\ue00d\ue005 无解。 Page 27满足的三个解向量b1\ue01a.方程组如果非齐次线性,且矩阵3是Ax设321,. 1\ue0080\ue003,\ue008\ue008\ue010\ue011ARm\ue005A\ue005\ue00b ,3221\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue005\ue002\ue008\ue008,1132\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue00f\ue005\ue002\ue008\ue008\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue003\ue005\ue00210113\ue008\ue008的通解b求Ax \ue005 Page 28, 1)(,3 无关的解向量\ue005\ue00b的基础解系中含有ARm0A矩阵是解 \ue000.2,\ue00513\ue002\ue002个线性\ue005\ue003\ue005\ue006Ax令\ue008\ue002则\ue00f,,133221c)b1a\ue010\ue011\ue010\ue011\ue005\ue005\ue005\ue002,\ue01a\ue005\ue005\ue002\ue003\ue008\ue008\ue008\ue01a\ue01b2\ue01a\ue0083\ue010\ue010\ue003\ue008b231\ue003201)(2111\ue01a\ue019\ue010\ue00f\ue00f\ue005ca\ue008,23)(23\ue01a\ue01b\ue01a\ue019\ue010\ue005\ue003\ue002\ue005acb\ue008,25210(22\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue003cba\ue008 Page 29 ,21121\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue003\ue005\ue003\ue008\ue008\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue005\ue00323131\ue008\ue008.0的基础解系中的解向量的通解为故b1\ue010\ue011,为\ue005AxAx \ue005\ue010\ue0112为任意实数,212312312.112132\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue002\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue011\ue010\ue00f\ue002\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue00f\ue003\ue005\ue01a\ue01a\ue01b\ue01a\ue019\ue010\ue010\ue00fkkxxx1其中kk