- #1

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The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following:

(a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and $S=\{s_{\alpha}\mid \alpha\in \mathbb{R}\}$, where $d_{\alpha}=\begin{pmatrix}\cos (\alpha) & -\sin (\alpha) \\ \sin (\alpha ) & \cos (\alpha )\end{pmatrix}$ and $s_{\alpha}=\begin{pmatrix} \cos (\alpha )& \sin (\alpha ) \\ \sin (\alpha) & -\cos(\alpha)\end{pmatrix}$.

(b) For all $\alpha\in \mathbb{R}$ is $B_{\alpha}$ an orthonormal basis of $\mathbb{R}^2$. It holds that $B_{\alpha}=(e_{\alpha}, f_{\alpha})$, where $e_{\alpha}\begin{pmatrix}\cos \left (\frac{\alpha}{2}\right ) \\ \sin \left (\frac{\alpha}{2}\right )\end{pmatrix}$ and $f_{\alpha}\begin{pmatrix}-\sin \left (\frac{\alpha}{2}\right ) \\ \cos \left (\frac{\alpha}{2}\right )\end{pmatrix}$

(c) Calculate $M_{B_{\alpha}}(\sigma_{\alpha})$,where $\sigma_{\alpha}(x)=s_{\alpha}x$.

I have done the following :

(a) To show that $D\cap S=\emptyset$, we assume that this is not true, i.e. that there is a matrix that belongs to $D$ and to $S$. Then for some $\alpha\in \mathbb{R}$ it must hold that $-\sin (\alpha)= \sin (\alpha) \Rightarrow \sin (\alpha)=0$ and that $\cos (\alpha)= -\cos (\alpha) \Rightarrow \cos (\alpha)=0$. There is no such $\alpha$ and therefore the intersection is empty.

How can we show that $O(2, \mathbb{R})=D\cup S$ ?

(b) We have to show that $e_{\alpha}$ and $f_{\alpha}$ are linearly independent, so that we can say that $B_{\alpha}$ is a basis of $\mathbb{R}^2$, right? To show also that itis an orthonormal basis, we have to show that the vectors $e_{\alpha}$ and $f_{\alpha}$ are orthogonal, i.e. their dot product is equal to $0$ and that it is normal, i.e. that both vectors have length $1$, right?

(c) Do we have to write the columns of $s_{\alpha}$ as a linear combination of $e_{\alpha}$ and $f_{\alpha}$ ?

:unsure: