(Solved): The standard basis S={e1,e2} and a custom basis B={b1,b2} for R2 are shown in the figur ...

The standard basis $\mathcal{S}=\left\{{\mathbf{e}}_1,{\mathbf{e}}_2\right\}$ and a custom basis $\mathcal{B}=\left\{{\mathbf{b}}_1,{\mathbf{b}}_2\right\}$ for ${\mathbb{R}}^2$ are shown in the figures below. a. 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 $?1,2?$. $\begin{array}{l}{\left[{\mathbf{b}}_1\right]}_{\mathcal{B}}=\\ {\left[{\mathbf{b}}_2\right]}_{\mathcal{B}}=\end{array}$ b. 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 $?1,2>$. $\begin{array}{l}{\left[{\mathbf{b}}_1\right]}_{\mathcal{S}}=\\ {\left[{\mathbf{b}}_2\right]}_{\mathcal{S}}=\end{array}$ c. Find the change of basis matrix from custom $\mathcal{B}$-coordinates to standard $\mathcal{S}$-coordinates. $[id{]}_{\mathcal{B}}^{\mathcal{S}}=[\stackrel{?}{?}]$ d. Find the change of basis matrix from standard $\mathcal{S}$-coordinates to custom $\mathcal{B}$-coordinates. $[id{]}_{\mathcal{S}}^{\mathcal{B}}=\left[\begin{array}{l}?\\ ?\end{array}\right]$ e. Find the $\mathcal{S}$-coordinates of the vector $?7{\mathbf{b}}_1+3{\mathbf{b}}_2$. Enter your answer as a vector of the form $?1,2?$. ${\left[?7{\mathbf{b}}_1+3{\mathbf{b}}_2\right]}_{\mathcal{S}}=$ f. Find the $\mathcal{B}$-coordinates of the vector $?4{\mathbf{e}}_1+9{\mathbf{e}}_2$. Enter your answer as a vector of the form $?1,2?$. ${\left[?4{\mathbf{e}}_1+9{\mathbf{e}}_2\right]}_{\mathcal{B}}=$