求一个正交变换把二次曲面的方程[tex=14.5x1.429]Q0cNgDEWV7zbKqubDhcBDNrsyunIxrFhuxNn8/b2ncwKufbraTBdm1fH3UvGS6+Q[/tex]化成标准方程,并指出该方程表示什么曲面?
解记二次曲面为[tex=2.071x1.214]0Aa4BLYb1CTBOTCQu7MCjA==[/tex]则f为二次型,它的矩阵为[tex=10.286x3.929]sSXBpxJWudVpH1R35o4LnCAE2Cwh8R50KEjQa2dPfa6T/3MZx8gVyByUe8fQfqPExW+r12z657tlnciu+MNe5pqSIi+Unh1fUlOCnqcIrh4dCB7aoN2w/PiHaEFb/+IY[/tex][tex=23.929x4.143]ee5DIWvmSq79Tvc0YR20LdYHq6Vtl3ao4vZaP0egxGXldWk8vsUIJdHEqvYiXV1ZzvwBtdOxCGkuIKdsqlsKDMknR/iozaECUYquMIgUKu+hWmPS2GdJ0GKxgZ+0qqqakrp6HU1hYl5g8BZzq1S30CQG8ZSJyDWNQN+6BbLBYetYcjZm+IEnTf5VTUPMjyJBe53Peh/IGJPY5hJLPqNda+f0o/rj5CAlT30LN/r7CIOUxsa/knh2pVbOIUTs4dJtKLLyWW3nCe2vLaufxojbVrRnOziDd4zFdIpJweQykalAd9Drolm1jqoq88BVBbBl[/tex][tex=18.643x3.929]Uyz5s0rmQIddjb5Jc2T/YaaAUuYLwh++dd9ZQ3IZ5bfTCDfYneQ1vcpPVgpkjlbHfBBxlBo0t6syo9eHBhznZeAr0j8HPXhY28hclk1tboDeexw6a+nK1wCdEGexFpDrr7KxOpmHGSC1oUhe9dN4x/WyPphw3r7pk8XPashkQMsWb+OfC0xJntwz7Wlq/BhPTPlfU8eLt1dN5k++gHGU8lt6KdSxWNi+DonDkYRgW+EnELnPosDZBTk2L2fTg5t0[/tex][tex=7.929x1.357]t3tDYzO+e3kLkvM5Jr+oJwUlTbXGhwabK/mgK+Vjeo7cgPkhw6gGtFL9EWarZcF7[/tex]所以A的特征值为[tex=8.857x1.214]sHJ8MVFGqADO2iHOA98xyJDa2CcvTOt2ni2zyeOsMlWKWu/gPYIrm2ZoQ9JGV/WDdrliw+Q8On0QPpzWE6rSZQ==[/tex]对应特征值[tex=2.214x1.214]VED7ZwYpqQgIj7DwVDPbFg==[/tex].解方程[tex=2.643x1.0]XUIPUDhcajktaIzleb/bxQ==[/tex]由[tex=10.286x3.929]sSXBpxJWudVpH1R35o4LnCAE2Cwh8R50KEjQa2dPfa6T/3MZx8gVyByUe8fQfqPExW+r12z657tlnciu+MNe5pqSIi+Unh1fUlOCnqcIrh4dCB7aoN2w/PiHaEFb/+IY[/tex][tex=17.357x3.857]075gCzZzsMRb6HYXYk9X95PG+rzYUNDK1ewWM9TUamAO7MOlHLzc5cZycYxGkYgFyVxZKRRPIGstFEsY5nHgFaC4VUyvaEIhqQn9Bke53PGlohFIxfQYhPg3kaSK42d+5GP/pOCX5Z6asfU2d/kZjB0W34J0mWTzrMG0Xg4+b3OpDXVKCameGbtm15CpLSTKSV5iNVcr1t1MrBdkFPO91NamIM5p8tjPjPdqlyD3228=[/tex][tex=15.643x3.643]075gCzZzsMRb6HYXYk9X95PG+rzYUNDK1ewWM9TUamD5pJWd7nMZ3mVViNwvZIzYl8fvH03jfaYDYIgJI4HvCTZSyCKCb/OCvOCIXbWmmXCcvD544BsQEcS9zbcuH9yputrh8liZLaE3E1M1d1ymeBBt2TitxBNJ2Yo7Prp9wbquKkYhZOHmEx4ZZY7B4lCe8XtGKZpUd3U5MuXTCPCzekrXdv9DUd6txuH5jjFBtKM=[/tex]得特征向量去[tex=5.286x3.5]+dFiBDKfihCShn3694A/Rok69NMhVEMMzbnnBgVPnmybGXt3UKdw5MyDL2RiUsHpGsn5IxD//XRlO8MZqXOxhA==[/tex]单位化得[tex=6.786x3.5]4IwCB7mUZT1bCJOXMr2ql/WD4Jr7xQa6wNu3u7bSc76xJ9rEuLxKXl7IFZ5tH/G9ElvzTt1Vx9PJCPzvNcX/aAJ89o29/lTKePSK6v78hikxL60OLQlrCVJgwbLlj5jF[/tex]对应特征值[tex=2.214x1.214]IMsqjeaLTZIrWNwLmqhWWw==[/tex]解方程[tex=5.714x1.357]tz155VRG27pO18Y5F4pHQQ==[/tex]由[tex=12.357x3.929]L8XovonL/2Bw0HJevRuDM/RYOTapW4yzBakuxYrzaaSHbe032Y5du1jF/MFyI0AcJukfbE8swcyFteDy3wVBqAheblTs/ZKqUZHQv/GfCwUfTwmohIv9phKselMKtUnY[/tex][tex=18.643x3.929]075gCzZzsMRb6HYXYk9X95TZAd/d3G7JYEaj1VE2AFIBQalYyB79GD4b9+8he0nHebHTNVn5w4YJYACowxbT0izl9984dDUS33IqzrZbE1qu4aH+VOqzs2c7MuyOKO3m1oOSOoz6a4loLVcpbCb6pZMcRGrQj01rXiUN+pAwPmrq+56qwcX6jUnn3RbGBrJvHr4fklI5Af4BKuodpFdT5GFBfQ7jic1DuklgKQXuli9CtPMlM9C45bgNLAp9X7T3[/tex]得特征向量[tex=6.071x3.643]SiRR31LQAQq/mHrORCY/t6mGjAma5RtRJ4A2XHxKrMDzG5ecmiyuWp5XnHpeqCsTWeOmEjHPUoiswBn4PLha+Q==[/tex]单位化得[tex=6.571x3.5]PV6WpOUCtKHVBPXKOWA3aMUzP3lEZnlwVCQj+i2S8FrGiKG0cjJiWUdvFtnqLbcFzPI0fDb9a4wsxp4b5YGRGe4UbLAQyHB2IUagyAP4SKw=[/tex]对旋特征值[tex=3.714x1.214]bG7nwGyI2D9YJTOlVf90qGd1qMxDSO3TvLObjheRmrg=[/tex]解方程[tex=6.214x1.357]PEC5s6o8S+Zds8UYRXlc+g==[/tex]由[tex=12.857x3.929]okimteR2HvI/hZK7urll9Sb5m1PhlmCY61xE7wHVYgyhApnO5b/UlWJ+ZImGUqYGtzRuSd35BbQje9LxLfWWbe2kQ1VxVjYLy2IuVeP5esdHqKdQEV4cGQ2cTjEJdW7e[/tex][tex=21.429x4.0]075gCzZzsMRb6HYXYk9X9ykCivDllLAwHAGTtBKeGnkyjef0b6mY1ZEAYpwSQZSVnTb6UU7LyGFkg8id+7XQ5suznknZXTA0B9pM8O3tQ+2hDqQ0fukp3hiZxjdJCptS6ulF8pO9VIrgJ4X0J60sAxF68snt3NfTPPEuofrx2YTuy2mSzaqJpa/R1TVchvnkhTBxzounLwHUjtR+wFBI2JbTtxX3Ns7Z7jaEGd2C2GYtweXcG1TelsPVA7oRGmQX[/tex][tex=26.286x4.786]075gCzZzsMRb6HYXYk9X9ykCivDllLAwHAGTtBKeGnmtIT0y8d7g0qSHIgPK79W5dTZiiBB4x1emtU0IE1Cc52PnxD14SX9cE7DM5hOdw29Aco7TkHs8AnPm2eeZHAziuqY/MCZ5nynQoxIcB1I4UbzS2OzawAQEiKNrcNrYOBt5H8l24mMguN1mZYRVv8djKvV/eyLdTdeB2GkDkJTXlOh6l55DHuwHkIs4KlMJSSJMU0jP9RJj35u/4WDPb+WtGH3WUeKnAjki/loHTviIqgdtBCg1JmA6q8mCRzBctFEX9XIAhfeKQFXlBV98qCAq8oeIPc3FMBuKacih7M33YzayVU4ZGyHvXA9EravqQDEr38TnYAf6gqiLXvpd9AfU[/tex]得特征向量[tex=5.643x3.643]yxyCRYeLij2awB0kzSi+gHfl7IrL9OHdHxCOqbRMmIiRh+ss6Wm8MJMOQVJCAWaY8RBUA/v4Vi0icB3lI6DpVg==[/tex]单位化得[tex=6.571x3.643]3VNwycHvMk1uEMCnO3Zj6wDMlCjDoDtFSJ0F90PQhs7SV4eQnBpTFDTSgA24sYW2Dpp9RhMeC8v2kViKK3v6tjPjx7WZW1aoRZ83ZNRLxxI=[/tex]令[tex=6.929x1.357]puDx9TRZShh+LZ3I5Cr6epuBgNEFF4tiVbQmzIF8gH7rAF+C9EBZtAnEHE9o317AtRB6z6t9HA9KSq6t3v7+dGWATMqWCGNzLNLkgffgQQD+nZvbAQudM+fW48SHIy0EazbVZuNuZvkyfUJgJXD14w==[/tex],则P是正交阵,作正交变换[tex=2.786x1.214]XyCcuT0aUDZ7FjbGEsDfcw==[/tex]即[tex=16.571x8.357]075gCzZzsMRb6HYXYk9X96r0kWKSapX3uFTiObPnpwgbpgDJjGmwyFhEy1SS/CpGAA5WPrOdS4n0tG7SBPaf65WaZ5S5T0q65NAup9Mt3adJlWxrOZK9AiP9ohDwB28mMsuKUDvfObgMEaSRKayhcXuYLPgMGEpLDFku4MDFmNBb5/zuySDNWXVuth9nluBqEDynieEOECeW8tSppMcWcVkVs1DJa3mb0cyng8eYOyHZAwDRHo1fa4xDAjF1KG64V43IFFj0V3zswwkBUnQ18mPWIR8vaXDJ4OnwTQWaL+/S5W8iltbK7U5tQ3Nx3RzE3KqFEOuiUgNgOUrjuKCFnq9FPJHS16d43VTTJ6a5F5k=[/tex]在此变换下.二次曲面的方程化为标准方程[tex=6.571x1.429]QuLxWyrITws0A1SS7wwE3oMg/OZ5Et9BIzLFWGSaBP8=[/tex]它表示椭圈柱面.
举一反三
- 求一个正交变换把二次曲面的方程 [tex=7.429x1.286]6bEpj2XYV2Y9+0UUtctIFMzyHxa86SJYraduW99laS4=[/tex][tex=9.214x1.286]yRqLYw/JCULXechdAGayMn8EOMoJfqv0I594akrQ+5g=[/tex]化成标准方程.
- 求一个正交变换[tex=8.143x3.357]jcCMHflCR8OS9TosV6N5vBJLPA1uBQTxcynNa8WyQjcsEDIpJ03M0BX+8NK/C3EVtj3hFZwtZLEqfpWEBPgb9bqS7Ui18ZKUcUb7W7LFhJFfQVW2Hnc9/Y+y63FclgXj8WP2VHFOREMkIinBpyi66g==[/tex],将二次曲面方程[tex=13.0x1.429]7PPpgWZDCTQdo4vyO8xzv6JyN1U5MOyW2LUmbFrh/zjJwEDOKkZtlXI/lN+8JJGa[/tex]化为标准形方程,并问该二次曲面是什么类型的曲面.
- 作直角坐标变换,把下述二次曲面方程化成标准方程,并且指出它是什么二次曲面?[tex=11.214x1.429]eBimLScnrNmmjfOZUa8NiiCmqF6EK2fOGkzvWk0lmjGb2YgN46GpIcx/Bt8Xx0Pf[/tex]。(3)
- 作直角坐标变换,把下述二次曲面的方程化成标准方程,并且指出它是什么二次曲面:[tex=4.929x1.429]Bc2b0DeKCNVTXfu67RNsuIzYIESUAS4SyIUrdDWJrSo=[/tex][tex=7.714x1.357]AG1WGzF5US30Oj42l4nceK83/F+2L3lR3U6fPfW/Fz8=[/tex].
- 利用坐标变换使方程[tex=3.5x1.429]udtIIahB5iVp7t7QwgrZ2g==[/tex] 中减少一个一次项,化成标准方程,指出是何种曲面.
内容
- 0
利用不变量判断下列二次曲面为何种曲面,并求出它的简化方程与标准方程:[tex=9.571x1.429]gZK8OvK5tqXhyNuyTNgtr1oR1Q+Hb0o1obDbrZOjIeU=[/tex].
- 1
利用不变量判断下列二次曲面为何种曲面,并求出它的简化方程与标准方程:[tex=14.0x1.429]BY+chx8tXev6KOK3vf0gZprVGZykVLy81bPIcpmnXELbAJdxsLxLY4PsqMiBsebw[/tex].
- 2
利用不变量判断下列二次曲面为何种曲面,并求出它的简化方程与标准方程:[tex=12.786x1.429]4FMDVPLuD57GDhXGjCa6CNsAzhXAgh8W458qnKZIQjf+yTS1HJuFaPHJCLVO8FKg[/tex]
- 3
指出下列各方程表示哪种曲面:(7)[tex=5.286x1.429]WdyGWxTsN/IN7z6b8y8iSQ==[/tex]
- 4
指出下列各方程表示哪种曲面:(2)[tex=5.714x1.429]54QY+hvUN5WO91XfeF0VEFrVAE89ubfGMSRno85pBMk=[/tex]