Seien \( \mathcal{B}=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\} \) die Standardbasis für \( \mathbb{R}^{2 \times 2} \),$$ \mathcal{B}^{\prime}=\left\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right)\right\} $$eine weitere Basis für \( \mathbb{R}^{2 \times 2} \) und \( T: \mathbb{R}^{2 \times 2} \rightarrow \mathbb{R}^{2 \times 2} \) die lineare Abbildung mit$$ M_{\mathcal{B}}^{\mathcal{B}}(T)=\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right) . $$Berechnen Sie \( M_{\mathcal{B}^{\prime}}^{\mathcal{B}^{\prime}}(T) \).
