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Eir veur c hat beaoglich eirer gegesten Besis \( \left.B=8 a_{1}, a_{2}, a_{3}\right\}^{Z} \) des \( \mathbb{R}^{3} \) die bastellut \( C_{1 B}=(2,2,1)^{\top} \) zeigh ste, dess ach de vevuren \( b_{1}=a_{1}-a_{3}^{1 B}, b_{2}=a_{2}+a_{2}, b_{3}=a_{1}-a_{2}+a_{3} \) cire Basis des \( \mathbb{R}^{3} \) bíden. Rerectren se deve kardinater des veubrs a besigetich de- Basts \( \left.B^{\prime}=\left\{b_{1},\right\}_{2}, b^{\prime}\right\} \)
1) Spen; 3 veltoren und \( \mathbb{R}^{3} \) aso \( V \)
2) Lineervebthoyflet ;
Als けa T
Korcirefer: \( \left(\begin{array}{ccc}1 & 1 & 1 \\ 3 & 1 & -1 \\ -1 & 0 & 1\end{array}\right), c=\left(\begin{array}{l}2 \\ 2 \\ 1\end{array}\right) \)
\( \begin{array}{l} c_{1}+c_{2}+c_{3}=2 \\ c_{2}-c_{1}=2 \\ C_{3} \cdot C_{1}=1 \\ \rightarrow\left(\begin{array}{ccc:c} 1 & 1 & 1 & 2 \\ 0 & 2 & -1 & 2 \\ -1 & 0 & 1 & 1 \end{array}\right) \xrightarrow{R_{1}+l_{j}+k_{2}}\left(\begin{array}{cccc} 1 & 1 & 1 & 2 \\ 0 & 1 & -1 & 2 \\ 0 & 1 & 1 & 1 \end{array}\right) \\ \end{array} \)
\( \xrightarrow{C_{2} A_{3}+R}\left(\begin{array}{ccccc}1 & 1 & 1 & 1 & 2 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 2 & 1 & 1\end{array}\right) \)
\( \begin{array}{l} 2 c_{3}=1 \rightarrow c_{3}=1 / 2 \\ c_{2}-c_{3}=2+c_{2}=9 / 2 \\ c_{1}+c_{2}+c_{1}=+c_{1}=-1 \end{array} \)
\( l=\left(\begin{array}{c} -1 \\ 5 / 2 \\ 1 / 2 \end{array}\right) \)
Ist das richtig ?
