高等代数(北大版)第5章习题参考答案.doc

YOUR LOGO 原 创 文 档 请 勿 盗 版第五章二次型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）4x1 x22x1 x32x2 x3 ；2222）x12 x1 x2 2 x2 4x2 x34x3；223）x13x22x1 x22x1 x36x2 x3 ；4） 8x1 x42x3 x42x2 x38x2 x4 ；x1 x2x1 x3x1 x4x2 x3x2 x4x3 x4 ；5）2226）x12x2x44x1 x24x1 x32x1 x42x2 x32x2 x42x3 x4；22227）x1x2x3x42x1x22x2 x32 x3 x4 ；解 1 ）已知f x1 , x2 , x34 x1 x22 x1 x32x2 x3，先作非退化线性替换x1x2 x3y1y1 y3y2y2（ 1）则22f x1 , x2 , x34 y14y24 y1 y322224y14 y1 y3y3y3 4y23222 y1y3y34 y2 ，再作非退化线性替换1212y1z1z3（ 2）y2y3z2z3则原二次型地标准形为222f x1 , x2 , x3z14z2z3 ，最后将（ 2）代入（ 1），可得非退化线性替换为精品学习资料——勤奋，为踏入成功之门地阶梯第 1 页，共 41 页1212z31212x1z1z2z3x2z1z2z3（ 3）x3于为相应地替换矩阵为1212012121120012010010110110001T10，且有100040001T AT；2222 ）已知f x1 , x2 , x3x12x1 x22x24x2 x34 x3 ，由配方法可得2222f x1 , x2 , x3x12x1x2x2x24x2 x34x322x1x2x22x3，于为可令y1y2y3x1x2 x3x22 x3，则原二次型地标准形为22fx1 , x2 , x3y1y2，且非退化线性替换为x1x2x3y1y2y3y22 y32 y3，相应地替换矩阵为100110221T，精品学习资料——勤奋，为踏入成功之门地阶梯第 2 页，共 41 页且有112012001110122024100110221100010000T AT；22（ 3）已知f x1 , x2 , x3x13x22 x1 x22 x1 x36x2 x3 ，由配方法可得22222fx1 , x2 , x3x12x1 x22x1 x32x2 x3x2x34x24x2 x3x322x1x2x32x2 x3，于为可令y1y2y3x12x2x3x2x3x3，则原二次型地标准形为22fx1 , x2 , x3y1y2 ，且非退化线性替换为1232x1y1y2y312y312x2y2y3，x3相应地替换矩阵为12120321211T00，且有12120321211112320121200111133130100010000T AT00；1（ 4）已知f x1 , x2 , x3 , x48x1 x22x3 x42x2 x38x2 x4 ，精品学习资料——勤奋，为踏入成功之门地阶梯第 3 页，共 41 页先作非退化线性替换x1x2 x3 x4y1y2 y3 y4y4，则2fx1 , x2 , x3 , x48y1 y48y42 y3 y4 2y2 y38y2 y421 y1 y21 y812121 y8248y2 yyy4123123221212188y1y2y32 y2 y322121218148y1y2y3y42y1y2y32 y2 y3 ，再作非退化线性替换y1y2y3y4z1z2 z2 z4z3z3，则221 z5 z3 z5 z43 zfx , x , x , x8z2z1 2 3 412341232884222z22z3 ，再令5434w1z1x2x3w2w3z2z3，125838w4z1z2z3z4则原二次型地标准形为2222f x1 , x2 , x3 , x42w12w22w38w4 ，且非退化线性替换为精品学习资料——勤奋，为踏入成功之门地阶梯第 4 页，共 41 页1254w3w334x1w1w2w3w4x2x3w2w212，x4w1w4相应地替换矩阵为12001254110341101001T，且有2020000200008000T AT；（ 5）已知f x1 , x2 , x3 , x4x1 x2x1 x3x1 x4x2 x3x2 x4x3 x4 ，先作非退化线性替换x1x2 x3 x42 y1y2y3y4y2，则2f x1 , x2 , x3 , x42 y1 y2y22 y1 y32 y2 y32 y1 y42y2 y4 y3 y421234222y1y2y3y4y3y4y4y1，再作非退化线性替换z1z2y1y1y2y3y4，12z3y3y4z4y4即精品学习资料——勤奋，为踏入成功之门地阶梯第 5 页，共 41 页y1z112y2z1 z2z3z4，12y3z3z4y4z4则原二次型地标准形为342222f x1 , x2 , x3 , x4且非退化线性替换为z1z2z3z4 ，12x1z1z2z3z412x2z1z2z3z4，12x3z3z4x4z4相应地替换矩阵为1212121111111T，000010且有10000100001000034T AT；222（ 6）已知fx1 , x2 , x3 , x4x12 x2x44x1 x24x1 x32x1x42x2 x32x2 x42x3 x4 ，由配方法可得22f x1 , x2 , x3 , x4x1 2x1 2x22x3x42x22x3x42222x2 2x3x42x2x42x2 x32 x2 x42x3 x4精品学习资料——勤奋，为踏入成功之门地阶梯第 6 页，共 41 页23 x21 x21222，x2x2xx2 xxx123423434于为可令y1x12x22x3x432x412y2x2x3x4，y3y4x3x4则原二次型地标准形为12222fy12 y2y3 ，且非退化线性替换为x1y12y2y3y432y4x2y2y3y4，x3x4y3y4故替换矩阵为10002100132101111T，且有10000200001200000T AT；2222（ 7）已知f x1 , x2 , x3 , x4x1x2x3x42x1 x22x2 x32x3 x4 ，由配方法可得222f x1 , x2 , x3 , x4x22x2x1x3x1x32 x1 x32x3 x4x42222x1x2x32 x1 x3x32x3 x4x4x322222x1x2x3x3x42x1 x3x3x1x12222x1x1x2x3x3x4x1x3，于为可令精品学习资料——勤奋，为踏入成功之门地阶梯第 7 页，共 41 页y1y2y3y4x1x1 x3x1x2。