Hallo!
Könnte mir jemand eine Rückmeldung geben, ob meine Berechnungen richtig sind?
Aufgabe: Bestimme jeweils eine OB für folgende lineare Hüllen
i. \( \left\langle\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{c}0 \\ 1 \\ -1 \\ 1\end{array}\right)\right\rangle \)
ii. \( \left\langle\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)\right\rangle_{\mathbb{R}} \)
Problem/Ansatz:
(2) \( \left\langle\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{c}0 \\ 1 \\ -1 \\ 1\end{array}\right)\right\rangle_{\mathbb{R}} \)
\( u_{1}=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right) \)
\( u_{2}=\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right)-\frac{\left\langle\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right)\right.}{\left\langle\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1\end{array}\right)\right\rangle} \cdot\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right)= \)
\( \left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right)-\frac{1}{3}\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 0\end{array}\right)=\frac{1}{3}\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)-\frac{1}{3}\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right)=\frac{1}{3}\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)=0 \)
\( u_{4}=\left(\begin{array}{l}2 \\ 1 \\ 1 \\ 1\end{array}\right)-\frac{4}{3}\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)-0=\frac{1}{3}\left(\begin{array}{l}6 \\ 3 \\ 3 \\ 3\end{array}\right)-\frac{1}{3}\left(\begin{array}{l}4 \\ 4 \\ 0 \\ 4\end{array}\right)=\frac{1}{3}\left(\begin{array}{c}2 \\ -1 \\ 3 \\ -1\end{array}\right) \)
\( =\left(\begin{array}{r}0 \\ -1 \\ 1\end{array}\right)-\frac{\left\langle\left(\begin{array}{c}0 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)\right\rangle}{\left\langle\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right)\right\rangle} \cdot\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right)-\frac{\left\langle\left(\begin{array}{c}0 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)\right\rangle}{\left\langle\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)\right\rangle} \cdot\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)-\frac{\left\langle\left(\begin{array}{c}0 \\ 1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{c}2 \\ -1 \\ 3 \\ -1\end{array}\right)\right\rangle}{\left\langle\left(\begin{array}{c}2 \\ -1 \\ 3 \\ -1\end{array}\right),\left(\begin{array}{c}2 \\ -1 \\ 3 \\ -1\end{array}\right)\right\rangle} \cdot\left(\begin{array}{c}2 \\ -1 \\ 3 \\ -1\end{array}\right) \)
\( =\left(\begin{array}{c}0 \\ 1 \\ -1 \\ 1\end{array}\right)-\frac{2}{3}\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right)-0-\frac{(-5)}{15}\left(\begin{array}{c}2 \\ -1 \\ 3 \\ -1\end{array}\right)=\frac{1}{3}\left(\begin{array}{c}0 \\ 1 \\ -1 \\ 1\end{array}\right)-\frac{2}{3}\left(\begin{array}{l}1 \\ 1 \\ 0 \\ 1\end{array}\right)= \)
\( =\frac{1}{3}\left(\begin{array}{c}0 \\ 1 \\ -1 \\ 1\end{array}\right)-\frac{1}{3}\left(\begin{array}{l}2 \\ 2 \\ 0 \\ 2\end{array}\right)= \)
\( =\frac{1}{3}\left(\begin{array}{l}-2 \\ -1 \\ -1 \\ -1\end{array}\right) \hat{=}\left(\begin{array}{l}-2 \\ -1 \\ -1 \\ -1\end{array}\right) \)
\( \left\langle\left(\begin{array}{c}-1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)\right\rangle_{\mathbb{R}} \)
\( u_{1}=\left(-\frac{1}{-1}\right) \)
\( u_{2}=\left(\begin{array}{l}0 \\ 1\end{array}\right)-\frac{\left\langle\left(\begin{array}{l}0 \\ 1\end{array}\right),\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right)\right\rangle}{\left\langle\left(\begin{array}{c}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{c}1 \\ 1 \\ 1\end{array}\right)\right\rangle}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right)=\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)-\frac{0}{3}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right)=\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right) \)
\( u_{s}=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)-\frac{\left\langle\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right)\right\rangle}{\left\langle\left(\begin{array}{c}1 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ 1\end{array}\right)\right\rangle} \cdot\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right)-\frac{\left\langle\left(\begin{array}{l}1 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)\right\rangle}{\left\langle\left(\begin{array}{l}0 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)\right\rangle} \cdot\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)= \)
\( =\left(\begin{array}{l}1 \\ 0\end{array}\right)-\frac{0}{3} \cdot\left(\begin{array}{c}-1 \\ 1 \\ 1\end{array}\right)-\frac{1}{2}\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}2 \\ 2 \\ 0\end{array}\right)-\frac{1}{2} \cdot\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right) \)
\( O B=\left\{\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right),\left(\begin{array}{c}2 \\ 1 \\ -1\end{array}\right)\right\} \)
