高等代数(北大版)第5章习题参考答案[1].docx

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1、第五章二次型1．用非退化线性替换化下列二次型为标准形，并利用矩阵验算所得结果。1）4x1x22x1x32x2x3；2）x122x1x22x224x2x34x32；3）x123x222x1x22x1x36x2x3；4）8x1x42x3x42x2x38x2x4；5）x1x2x1x3x1x4x2x3x2x4x3x4；6）x122x22x424x1x24x1x32x1x42x2x32x2x42x3x4；7）x12x22x32x422x1x22x2x32x3x4。解1）已知fx1,x2,x34x1x22x1x32x2x3，先作非退化线性替换

2、x1y1y2x2y1y2（1）x3y3则fx1,x2,x34y124y224y1y34y124y1y3y32y324y222y1y33y324y22，再作非退化线性替换y11z11z322y2z2（2）y3z3则原二次型的标准形为fx,x2,x3z24z2z2，1123最后将（2）代入（1），可得非退化线性替换为x11z1z21z322x21z21（3）z1z322x3z3于是相应的替换矩阵为1011011102222T110111100，00100120201且有100TAT040。0012）已知fx1,x2,x3x122x1x

3、22x224x2x34x32，由配方法可得fx1,x2,x3x122x1x2x22x224x2x34x32x1x2x22x2，23于是可令y1x1x2y2x22x3，y3x3则原二次型的标准形为fx1,x2,x3y12y22，且非退化线性替换为x1y1y22y3x2y22y3，x3y3相应的替换矩阵为112T012，001且有100110112100TAT110122012010。221024001000（3）已知fx1,x2,x3x123x222x1x22x1x36x2x3，由配方法可得fx,x,x3x22xx22xx32xxx

4、2x24x24x2xx2121112323233x1x2x322x2x32，于是可令y1x1x2x3y22x2x3，y3x3则原二次型的标准形为fx1,x2,x3y12y22，且非退化线性替换为x1y11y23y322x21y21y3，22x3y3相应的替换矩阵为1132211T0，22001且有11310011122100110133011010。TAT222213000003110122（4）已知fx1,x2,x3,x48x1x22x3x42x2x38x2x4，先作非退化线性替换x1y1y4x2y2，x3y3x4y4则fx1

5、,x2,x3,x48y1y48y422y3y42y2y38y2y428y422y41y11y21y31y11y21y322822881y11y21y322y2y322881y11y21y321y32y42y1y22y2y3，2284再作非退化线性替换y1z1y2z2z3，y3z2z3y4z4则81z15z23z325z23z32fx1,x2,x3,x4z42z1288442z222z32，再令w1z15x23x344w2z2，w3z3w41z15z23z3z4288则原二次型的标准形为fx1,x2,x3,x42w122w222w3

6、28w42，且非退化线性替换为x11w15w23w3w4244x2w2w3，x3w2w3x41w1w42相应的替换矩阵为15312440011T11，0010012且有20000200TAT02。000008（5）已知fx1,x2,x3,x4x1x2x1x3x1x4x2x3x2x4x3x4，先作非退化线性替换x12y1y2x2y2，x3y3x4y4则fx1,x2,x3,x42y1y2y222y1y32y2y32y1y42y2y4y3y423y42y1y2y3y42y31y4y12，24再作非退化线性替换z1y1z2y1y2y3y4

7、z3y31y4，2z4y4即y1z1y2z1z2z31z42，1y3z3z42y4z4则原二次型的标准形为fx1,x2,x3,x4z12z22z323z42，4且非退化线性替换为x1z1z2z31z42x2z1z2z31z4，2x3z31z4x4z42相应的替换矩阵为11112T11112，001120001且有1000TAT01000010。00034（6）已知fx1,x2,x3,x4x122x22x424x1x24x1x32x1x42x2x32x2x42x3x4，由配方法可得fx1,x2,x3,x4x22x2x22x3x2x2

8、2xx2114342x22x3x422x22x422x2x32x2x42x3x43x31x42x12x22x3x422x21x3x42，222于是可令y1x12x22x3x4y2x23x31x4，22y3x3x4y4x4则原二次型的标准形为fy12