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[原创]问题解答43

1. 高等代数问题解答43

Example 1. %
设$V$是$n$维欧氏空间,$V_{1},V_{2}$是$V$的两个$m(0<m<n)$维子空间.证明:存在$V$的正交变换$\tau$使得
$\tau(V_{1})=V_{2}.$

\textbf{证明:}设$V_{1}$与$V_{2}$的标准正交基分别为
$$\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$$

$$\beta_{1},\beta_{2},\cdots,\beta_{m},$$
将其分别扩充为$V$的标准正交基为
$$\alpha_{1},\alpha_{2},\cdots,\alpha_{m},\alpha_{m+1},\cdots,\alpha_{n},$$

$$\beta_{1},\beta_{2},\cdots,\beta_{m},\beta_{m+1},\cdots,\beta_{n}.$$

$$\tau(\alpha_{i})=\beta_{i},i=1,2,\cdots,n,$$
则$\tau$是$V$的正交变换,且
$$\tau(V_{1})=\tau(L(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}))=L(\tau(\alpha_{1}),\tau(\alpha_{2}),\cdots,\tau(\alpha_{m}))=L(\beta_{1},\beta_{2},\cdots,\beta_{m})=V_{2}.$$

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