Review Note

Last Update: 08/28/2024 07:49 AM

Current Deck: LA 2

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Was ist eine Drehung?
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für \(\varphi \in \mathbb R\) ist \(D_{\varphi}: \mathbb R ^2 -> \mathbb R^2\)
\(D_{\varphi}(\begin{pmatrix} x_1 \\ x_2\end{pmatrix})= \left( \begin{array}{rrr} cos(\varphi) & -sin(\varphi) \\ sin(\varphi) & cos(\varphi) \\ \end{array}\right)*\begin{pmatrix} x_1 \\ x_2\end{pmatrix}\) ist eine Isometrie 

\(D_{\frac{\pi}{2}}(x)=\left( \begin{array}{rrr} cos(\frac{\pi}{2}) & -sin(\frac{\pi}{2}) \\ sin(\frac{\pi}{2}) & cos(\frac{\pi}{2}) \\ \end{array}\right)*\begin{pmatrix} x_1 \\ x_2\end{pmatrix} = \left( \begin{array}{rrr} 0 & -1 \\ 1 & 0 \\ \end{array}\right)*\begin{pmatrix} x_1 \\ x_2\end{pmatrix}= \begin{pmatrix} -x_2 \\ x_1\end{pmatrix}\)

Analog: \(D_{\pi}(x)=-x\) , \(D_{\pi}\) hat Eigenwert \(-1\) aber \(\varphi \notin \pi_{/_{\mathbb Z}}\) hat( keine Vielfache) \(D_{\varphi}\) keine reelen Eigenwerte

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Drehung

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