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

1. What are the custom $\mathcal{B}$-coordinates of the basis vectors $\mathbf{b_1}$ and $\mathbf{b_2}$? Enter your answers as a vectors of the form <1,2>.
   $\lbrack \mathbf{b_1} \rbrack_\mathcal{B} =$
   $\lbrack \mathbf{b_2} \rbrack_\mathcal{B} =$
2. What are the standard $\mathcal{S}$-coordinates of the basis vectors $\mathbf{b_1}$ and $\mathbf{b_2}$? Enter your answers as a vectors of the form <1,2>.
   $\lbrack \mathbf{b_1} \rbrack_\mathcal{S} =$
   $\lbrack \mathbf{b_2} \rbrack_\mathcal{S} =$
3. 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.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
4. Find the change of basis matrix from standard $\mathcal{S}$-coordinates to custom $\mathcal{B}$-coordinates.
   $\lbrack id \rbrack_{\mathcal{S}}^{\mathcal{B}} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
5. Find the $\mathcal{S}$-coordinates of the vector ${3} \mathbf{b_1} + {6} \mathbf{b_2}$. Enter your answer as a vector of the form <1,2>.
   $\lbrack {3} \mathbf{b_1} + {6} \mathbf{b_2} \rbrack_\mathcal{S} =$
6. Find the $\mathcal{B}$-coordinates of the vector ${-6} \mathbf{e_1} + {8} \mathbf{e_2}$. Enter your answer as a vector of the form <1,2>.
   $\lbrack {-6} \mathbf{e_1} + {8} \mathbf{e_2} \rbrack_\mathcal{B} =$