(Solved): Consider a basis \( \mathcal{B}=\left\{\left[\begin{array}{c}1 \\ -2\end{array}\right],\left[\begi ...

Consider a basis \( \mathcal{B}=\left\{\left[\begin{array}{c}1 \\ -2\end{array}\right],\left[\begin{array}{c}5 \\ -6\end{array}\right]\right\} \) for \( \mathbb{R}^{2} \). Use the change-of-coordinates matrix from \( \mathcal{B} \) to the standard basis in \( \mathbb{R}^{2} \) to find the coordinate vector \( [\vec{x}]_{\mathcal{B}} \) of \( \vec{x}=\left[\begin{array}{l}4 \\ 0\end{array}\right] \) relative to \( \mathcal{B} \).