A-A+
[原创]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
$\tau(V_{1})=V_{2}.$
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
$$\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$$
与
$$\beta_{1},\beta_{2},\cdots,\beta_{m},$$
将其分别扩充为$V$的标准正交基为
$$\alpha_{1},\alpha_{2},\cdots,\alpha_{m},\alpha_{m+1},\cdots,\alpha_{n},$$
与
$$\beta_{1},\beta_{2},\cdots,\beta_{m},\beta_{m+1},\cdots,\beta_{n}.$$
令
$$\tau(\alpha_{i})=\beta_{i},i=1,2,\cdots,n,$$
则$\tau$是$V$的正交变换,且
$$\tau(V_{1})=\tau(L(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}))=L(\tau(\alpha_{1}),\tau(\alpha_{2}),\cdots,\tau(\alpha_{m}))=L(\beta_{1},\beta_{2},\cdots,\beta_{m})=V_{2}.$$
