Change of Basis

  1. Let \(\mathcal{B}=\left\{\mathbf{b}_1, \mathbf{b}_2\right\}\) and \(\mathcal{C}=\left\{\mathbf{c}_1, \mathbf{c}_2\right\}\) be bases for a vector space \(V\), and suppose \(\mathbf{b_1}=6\mathbf{c_1}-2\mathbf{c_2} \) and \(\mathbf{b_2}=9\mathbf{c_1}-4\mathbf{c_2}\).
    (i) Find the change-of-coordinates matrix from \(\mathcal{B}\) to \(\mathcal{C}\).
    (ii) Find \([\mathbf{x}]_{\mathcal{C}}\) for \(\mathbf{x}=-3\mathbf{b_1}+2\mathbf{b_2}.\)  solution


  2. Let \(\mathcal{B}=\left\{\mathbf{b}_1, \mathbf{b}_2\right\}\) and \(\mathcal{C}=\left\{\mathbf{c}_1, \mathbf{c}_2\right\}\) be bases for \(\mathbb{R}^2\). Find the change-of-coordinates matrix from \(\mathcal{B}\) to \(\mathcal{C}\) and the change-of-coordinates matrix from \(\mathcal{C}\) to \(\mathcal{B}\).   solution

    3. \(\mathbf{b}_1=\left[\begin{array}{l}7 \\ 5\end{array}\right], \, \mathbf{b}_2=\left[\begin{array}{l}-3 \\ -1\end{array}\right], \, \mathbf{c}_1=\left[\begin{array}{r}1 \\ -5\end{array}\right], \, \mathbf{c}_2=\left[\begin{array}{r}-2 \\ 2\end{array}\right]\)