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	慵懒 2023-2-11 19:50

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发表于 2022-10-25 15:36 |只看该作者 |倒序浏览

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第十章双线性函数与辛空间

习题精解

1、设V是数域P上的一个三维线性空间，file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps1.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps2.png，file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps3.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps4.png，file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps5.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps6.png是它的一组基，f是V上的一个线性函数，已知

f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps7.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps8.png+file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps9.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps10.png)=1,f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps11.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps12.png-2file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps13.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps14.png)=-1,f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps15.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps16.png+file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps17.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps18.png)=-3

求f (Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps19.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps20.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps21.png+Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps22.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps23.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps24.png+Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps25.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps26.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps27.png).

解因为f是V上线性函数，所以有

file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps28.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps29.png f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps30.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps31.png)＋ f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps32.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps33.png)=1

f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps34.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps35.png)-2 f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps36.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps37.png)=-1

f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps38.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps39.png)+f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps40.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps41.png)=-3

解此方程组可得

f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps42.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps43.png)＝4，f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps44.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps45.png)＝－7，f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps46.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps47.png)＝－3

于是

f (Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps48.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps49.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps50.png+Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps51.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps52.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps53.png+Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps54.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps55.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps56.png).＝Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps57.png f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps58.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps59.png)＋Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps60.png f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps61.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps62.png)＋Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps63.png f (file:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps64.pngfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps65.png)

＝4 Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps66.png－7 Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps67.png－3 Xfile:///C:/Users/BAOJIN~1/AppData/Local/Temp/ksohtml13360/wps68.png

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