求复数域上线性变换空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的线性变换[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的特征值与特征向量.已知[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在一组基下的矩阵为:[tex=7.071x2.786]sSXBpxJWudVpH1R35o4LnN0OSmn/R2gssTNClSbe/RxSWLG4hXxToC26Jkm+jbgx3zmrOwG7+WuS1V1+XF7emQ==[/tex]

2022-06-16

求复数域上线性变换空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的线性变换[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的特征值与特征向量.已知[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在一组基下的矩阵为:[tex=7.071x2.786]sSXBpxJWudVpH1R35o4LnN0OSmn/R2gssTNClSbe/RxSWLG4hXxToC26Jkm+jbgx3zmrOwG7+WuS1V1+XF7emQ==[/tex]

答案：

2)设[tex=0.786x1.0]3akNjptD8YqOes80TdtIxQ==[/tex]在给定基[tex=1.929x1.0]uqfxMixAiniFaFwzhN2vVTgBtKplMq07gy1Ot9nIbTuggLhnWFA6UQ3SkwSmLJ8d[/tex]下的矩阵为[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex],且当[tex=1.857x1.0]c5C77r1McNqxiugbGpm6pA==[/tex]时,有[tex=2.357x1.0]Fkx7pjAcvtErR4kUWxr3Ig==[/tex]所以[tex=11.143x2.786]MjlGk2VBF8Tk4Cdof/YMXucwSk2D3/TGqB8PnIHdBg/lsz1yesu5bvNOm8L6Y7QHBcfrtB3lPDT997Vuw8M79TiiWLiyiquhr0BkoACcfXPrkwu2zfznMPMYXkwHq9sbRoFAjnGZdb7X3L90E5QSNA==[/tex]故[tex=0.786x1.0]3akNjptD8YqOes80TdtIxQ==[/tex]的特征值为[tex=3.929x1.214]lI2jIdIeKZq2X8UI5kI3C1N5Oafi4VSfuqTlZiWA3/Q=[/tex]解方程组[tex=6.714x2.786]mxN/b9Bocsxyl/gJB8u4gAf2nnFaiuzBo1mEo5wyi7gaItFOqY7NItUY9a7/YKHFVcW9sbBdcWQULzdC3caOcVrNK+ld/bKhiP9BjW1jM+WESLsrCcCOWvMRZPakNYmh[/tex]它的基础解系为[tex=6.571x2.786]mQb6aGgoarIegV00k+NZUUcZ979i15F0lXKpF8EYNiuGbhqC2az7LVmKfBlkKk8ch2uvFpHABUfNQOm/J77lCU9v3Gf7KUBQWCKi/W746w5z/b3H8011eYcvkEOX82rBorF0J+UrV9hQx3U9chkRSw==[/tex]因此[tex=0.786x1.0]3akNjptD8YqOes80TdtIxQ==[/tex]的属于特征值 0 的两个线性无关特征向量为 [tex=5.357x1.214]as5zgnuN6ivkD83J+xUiIISBO0Kgssp+P4clgsRXh1Y3l8URclxoGU8l5LkxBefGbvYKvsJhxnhl4IYIefOTVg==[/tex]故 [tex=0.786x1.0]3akNjptD8YqOes80TdtIxQ==[/tex]以 [tex=0.643x1.0]H4OBEtaFUUM3k47UOjlnFw==[/tex]的任一非零向量为其特征向量. 当 [tex=2.357x1.286]NmWLUlTOILHDfw7uqfi4DQ==[/tex] 时[tex=19.357x2.786]8QqowDobcyfcPuRAUWtpLE2frIGnT7pRO6PKDl8+8PMEKaEFvcn5IQ3XGZEZ+3WZNgrIejMxgRNtbkCMC2TDFxTCftfICiI/LIwmUJ67XxllDLyGhl0dmAh7YWGE8K5TjlrOi+VjPP0L5VuNe4w6tsbtOw2ik0vnrn0YPXqlHOv0eJGP8TvzrGAoJgm0Z8mJ[/tex]故[tex=0.786x1.0]3akNjptD8YqOes80TdtIxQ==[/tex]的特征值为[tex=7.0x1.214]xDEGziAar/e3JvvR62J5XFFfuBdYrMukK4JJttPD56lEkbfsNUQaWUkJEaJWRPOx[/tex]当[tex=2.643x1.214]9QWE7segbFULUuCj22hFcwYrLakDyor/bu8gWKvHyOo=[/tex]时,方程组[tex=7.857x2.786]fnpmC2J6JmQBLyo5NmGAzxeum+VUtlvVOmbJl4mXFw0+BYTB3MjgBfo/Xe2EY01B7JP2iqg9A/7QNSqDiME55tI6j3KienidgOqe5mZnfscWO/FDwDzEcXwU3btvBMzP[/tex]故[tex=0.786x1.0]3UKvB+w607mbn/eWBx9vkQ==[/tex]的属于特征值[tex=1.714x1.143]IS8RV+CoM/E9okt1/Jlr6A==[/tex]的全部特征. 的基础解系为[tex=4.429x2.786]075gCzZzsMRb6HYXYk9X9x0lyxeXitx8OwalltmuyAwRNVge8RNtgpIkonhD/RxmkTv0ewNBPx6742vOxGOUZA==[/tex]向量为[tex=4.714x1.286]lYauz4bQlNgr1T/6JDMoGNxWtRjlA/oBGwHEWJXLMAY=[/tex]其中 [tex=4.357x1.214]IXuiEWt8eBgU8N4pda5IUGy5EPSa0SX6wFB9fY3oloeRVmAfv1iEZEC4XcOQj9Zb[/tex] 当[tex=3.429x1.214]geOfvmQ3b6zXmC8PdDgRu3QoOLbW684+S/wHC7FciI4=[/tex]方程组[tex=7.857x2.786]fnpmC2J6JmQBLyo5NmGAzxeum+VUtlvVOmbJl4mXFw0+BYTB3MjgBfo/Xe2EY01B7JP2iqg9A/7QNSqDiME55tI6j3KienidgOqe5mZnfscWO/FDwDzEcXwU3btvBMzP[/tex]的基础解系为 [tex=3.143x2.786]075gCzZzsMRb6HYXYk9X90tIKBkgIhODADGpFArbO9jtWSrfCtNEBHa2YMllOrBIYhdP9rh8Ah4JU881LSLaIA==[/tex],故[tex=0.786x1.0]3UKvB+w607mbn/eWBx9vkQ==[/tex]的属于特征值[tex=1.643x1.286]XmexKPJL6ycyx9bOPeUOBGo3wXIykax115zJPHUlY/U=[/tex]的全部特征向量为[tex=4.714x1.286]NhQZR142hKWEhpSu+Y3Oc2GdfslRsbN72dsx5rOxrSg=[/tex]其中[tex=4.357x1.214]QJ3ZnNcLHkjjG1TETM1fia74eGjpYnyW6JlEiyzjJ336dtUjtkyU0lZqovorVWWH[/tex]

举一反三

内容

0
设[tex=4.643x1.0]2bOOLS2qYWpFCMCkhOx7kWKBZXCHc0rkmUgF/O9obdwPxSggBAHYEkc4KmIt+owdgvolNqDVZJPv8y6xbkiCkQ==[/tex] 是四维线性空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的一组基，已知线性变换[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在这组基下的矩阵为[tex=9.714x4.786]dEdrC9SQsN/3Vx39SaFo4F4k4j2a4XW0+ki4qRfuccZ3acDq0FvL6o/bF+WQXPHLP+sqGWr3situWKRnWapkr5ed8utdPa1QDBnWmM4vMGRQAeNdtMkTuQmnXcxPCj9/o6UgHc6gwEhnkF/JDVCroXTvP7C5kUQ+7yYTMkDBfGg=[/tex]1）求[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的核与值域；2）在[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的核中选一组基，把它扩充成[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的一组基，并求[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在这组基下的矩阵；3）在[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的值域中选一组基，把它扩充成[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex] 的一组基，并求[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在这组基下的矩阵．
1
证明：如果线性空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的线性变换[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]以[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]中每个非零向量作为它的特征向量，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数乘变换．
2
[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级线性空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]上的线性变换．１）若[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的某组基下矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是某多项式[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]的伴侣阵，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．２）设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最高次的不变因子是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．
3
求复数域上线性变换空间 V 的线性变换 A 的特征值与特征向量.已知 A 在一组基下的矩阵为:[tex=9.357x3.643]sSXBpxJWudVpH1R35o4LnKsVTfbfyFZRTXTQjYIzAt2y9cU2kEbt/DazXKJP9l5t2Bt/oaIQ8qRX5JDJPdhKl/LySmdRRaInHdxSnsyzGua6N1xcrC2lqrLrmTCmLMjZ[/tex]
4
求复数域上线性变换空间 V 的线性变换 A 的特征值与特征向量.已知 A 在一组基下的矩阵为:[tex=10.143x3.643]sSXBpxJWudVpH1R35o4LnAealDUzhvCNlp4875cyY8lsOlKdUWqlnc/MpS2zEgdOxKxnnwzc0uPkwje3Vg4xdvKd24z78QwC3oTPp93PyBeRF/KlkPR2qo/jydeyHbCh[/tex]