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本帖最后由 hbghlyj 于 2022-6-9 12:05 编辑 在$\Bbb R^3$中,兩個經過原點的平面,存在一個關於原點的旋轉將其中一個變到另一個. 一般地, 我們有
Theorem 1.3.4. For each dimension $m$, the natural action of $O(n)$ on the set of $m$-dimensional vector subspaces of $\Bbb R^n$ is transitive.
Proof:
Let $V$ be an $m$-dimensional vector subspace of $\Bbb R^n$ with $m > 0$.
Identify $\Bbb R^m$ with the subspace of $\Bbb R^n$ spanned by the vectors $e_1,\dots,e_m$. It suffices to show that there is an $A$ in ${\rm O}(n)$ such that $A(\Bbb R^m) = V$.
Choose a basis $\{u_1,\dots,u_n\}$ of $\Bbb R^n$ such that $\{u_1,\dots,u_m\}$ is a basis of $V$. We now perform the Gram-Schmidt process on $\{u_1,\dots,u_n\}$. Let $w_1 = u_1/|u_1|$. Then $|w_1| = 1$. Next, let $v_2 = u_2 − (u_2 · w_1)w_1$. Then $v_2$ is nonzero, since $u_1$ and $u_2$ are linearly independent; moreover, $w_{1} \cdot v_{2}=w_{1} \cdot u_{2}-\left(u_{2} \cdot w_{1}\right)\left(w_{1} \cdot w_{1}\right)=0$
Now let
$\begin{aligned} w_{2} &=v_{2} /\left|v_{2}\right| \\ v_{3} &=u_{3}-\left(u_{3} \cdot w_{1}\right) w_{1}-\left(u_{3} \cdot w_{2}\right) w_{2} \\ w_{3} &=v_{3} /\left|v_{3}\right| \\ & \vdots \\ v_{n} &=u_{n}-\left(u_{n} \cdot w_{1}\right) w_{1}-\left(u_{n} \cdot w_{2}\right) w_{2}-\cdots-\left(u_{n} \cdot w_{n-1}\right) w_{n-1} \\ w_{n} &=v_{n} /\left|v_{n}\right| \end{aligned}$
Then $\{w_1,\dots, w_n\}$ is an orthonormal basis of $\Bbb R^n$ with $\{w_1,\dots, w_m\}$ a basis of $V$. Let $A$ be the $n×n$ matrix whose columns are $w_1,\dots,w_n$. Then $A$ is orthogonal by Theorem 1.3.3, and $A(\Bbb R^m) = V$.
Reference: GTM149 |
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