The standard basis $\mathcal{S} = \lbrace \mathbf{e_1}, \mathbf{e_2} \rbrace$ and two custom bases $\mathcal{B} = \lbrace \mathbf{b_1}, \mathbf{b_2} \rbrace$ and $\mathcal{C} = \lbrace \mathbf{c_1}, \mathbf{c_2} \rbrace$ for $\mathbb{R}^2$ are shown in the figures below.

1. Find the change of basis matrix from custom $\mathcal{B}$-coordinates to standard $\mathcal{S}$-coordinates.
   $\lbrack id \rbrack_{\mathcal{B}}^{\mathcal{S}} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
2. Find the change of basis matrix from custom $\mathcal{C}$-coordinates to standard $\mathcal{S}$-coordinates.
   $\lbrack id \rbrack_{\mathcal{C}}^{\mathcal{S}} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
3. Find the change of basis matrix from $\mathcal{B}$-coordinates to $\mathcal{C}$-coordinates.
   $\lbrack id \rbrack_{\mathcal{B}}^{\mathcal{C}} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
4. Find the change of basis matrix from $\mathcal{C}$-coordinates to $\mathcal{B}$-coordinates.
   $\lbrack id \rbrack_{\mathcal{C}}^{\mathcal{B}} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$