若在仿射变换[tex=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/tex] 下一条直线的象与其自身重合, 则称这条直线为[tex=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/tex] 的不变直线. 求下述仿射变换的不变直线:[tex=7.786x3.357]GE56u9QCDTqcLxZ66HADyluO9vFS2xRx4ZwY/EdzUnJ9tmmI9dy+VSzk+pdlx3mqUxko5MU7XH0ADmDU+R/coZIyHZbBLzv9RyL0Tg1UJ72tnm2SR7RytSMTzW8Z2vzx[/tex]

2022-07-27

若在仿射变换[tex=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/tex] 下一条直线的象与其自身重合, 则称这条直线为[tex=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/tex] 的不变直线. 求下述仿射变换的不变直线:[tex=7.786x3.357]GE56u9QCDTqcLxZ66HADyluO9vFS2xRx4ZwY/EdzUnJ9tmmI9dy+VSzk+pdlx3mqUxko5MU7XH0ADmDU+R/coZIyHZbBLzv9RyL0Tg1UJ72tnm2SR7RytSMTzW8Z2vzx[/tex]

答案：

解: 仿射变换把直线变成直线. 设不变直线为[tex=7.357x1.214]SRLKDs+jB8+GwtoiYNtLLuZ36LyCOAmChHHqF8sI4BA=[/tex] 若[tex=4.429x1.357]oLtdETsrxK7auADxAHWdN+O4f8qsZz98uFuX6Z5yDno=[/tex] 则[tex=8.643x1.429]3CDu7omM9SsMJ8a63Ne6Y60SHCxJD1Tq3Vct3/euMOVyiyhfT8A5CIH1abdHOifTYyDFttjv+UntS05BAZeuTsPRGGeyOc/Vb+O6u/t222o=[/tex]即[tex=6.929x1.357]VXXygmvOlDpe5l7fFNvKnjjSQS08B7XJ+1zvnS3IsaHK/kZ2wcVKIxglyFuS1fQR[/tex] 代入化简后得[tex=16.857x1.357]z3AWgZ14GuqFfWm3BV05DkRQk/gdVse9psYlfYV0yS8Q61wgzeCFcs1GtZ43IHaq[/tex]因为[tex=0.714x1.0]Hl8mr56J4t0Ek5ZoqbFYYg==[/tex]上的任意点都满足此方程, 说明这个方程也是 [tex=0.714x1.0]Hl8mr56J4t0Ek5ZoqbFYYg==[/tex] 的方程. 因此有[tex=15.071x2.429]qM9upFG4QpIuxqnqn2HlkE77urt/RysScX36bb1ozRoZNc+bqKOYI+uvIgdDelfRigjYCEyV2mGdxF8PZcgCWg==[/tex]解得[tex=1.857x1.0]BXn3ykwvGNyQjZ3apQXpkA==[/tex] 或 [tex=2.143x1.0]x6axgTQfTsPw8bm8m6qaIQ==[/tex]所以[tex=12.143x3.929]GE56u9QCDTqcLxZ66HADylx+lmjVS0siTsJt8072Z2JZtruAsuXCdQ08jZ7btm1K3fBA5J3CvsGtsVkSwWrNxKmFuoFI4M+mlsadIxxbG0suO4yxfF7nAU5UrLFdHtFubIHJikmhvkWX4wrjyFgP6tzNAiBK9UyTdvKZT83a+l8yzvXtuBLSueuC/zQ1HNME[/tex]故不变直线有两条:[tex=4.429x1.214]VKY6IbQwGLybvBzHRwepOw==[/tex]和[tex=6.5x1.214]Z1Z+mz8ldS/FL+2Xh7HubwCBGrOh6nsx2ldTsbiQE0s=[/tex]

举一反三

内容

0
求下列仿射变换的不动点及不变直线:[tex=7.5x3.357]7EJHVCtO2IWq3KpdB+jQsrB6api6XTiW4A20y5aW7YzGpv1QRDZCIGesjv25JBoAbvqNAsKBkwvk7SmHtZ5Ky3SHZnAuIyutf2nO1eqjvFpn3Dil6MkY/RGrEhmZsukV[/tex]
1
证明:如果线性空间 [tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex] 的线性变换 [tex=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/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=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/tex] 是数乘变换
2
求下列仿射变换的不动点及不变直线:[tex=7.0x3.357]7EJHVCtO2IWq3KpdB+jQsrB6api6XTiW4A20y5aW7YxEEdwuyO5+A0yDwst+8ykMZHvpjmNgm2xQCbPDnp+99lDbast/JZQ/nog2vdUzjPjbYz0kbAz4EzNTIgb7Of+y[/tex]
3
证明:可逆变换是双射. 设[tex=0.857x1.0]xs/zPwdLSSAmQIIfXPkuWQ==[/tex]为可逆变换，即有可逆变换[tex=1.786x1.214]umkzA/lGqaggQDM+nhhh09myJYRUu5/6pEAdVHfe+CM=[/tex]使[tex=2.571x1.214]LWkxwTYD48E4tDCt0RyhtYBcWtJpQyTduF1EE2VRCUU=[/tex][tex=0.786x0.643]NhuTNiqjImitwKaHFutGOg==[/tex][tex=3.929x1.214]umkzA/lGqaggQDM+nhhh07H/+5I1WpXNDVJblzSP5Wt8zOi8EQ1Ya3jxpWR9hlqE[/tex]
4
求下述仿射变换的不动点:[tex=7.286x3.357]GE56u9QCDTqcLxZ66HADyluO9vFS2xRx4ZwY/EdzUnJZgvZkGZZXmH+1HiE5taZlWl5IEyDq6ne7DLA2KkubxBwQ8KxY7ngpjTC14uGBs8KyuNkHjgqIXklnFsK66cgy[/tex]