(Solved): Matrix representation of a linear transformation Recall the standard basis E={e1,e2} in R ...

Matrix representation of a linear transformation Recall the standard basis $E=\left\{{\vec{e}}_1,{\vec{e}}_2\right\}$ in ${\mathbb{R}}^2$ where ${\vec{e}}_1=\left(\begin{array}{l}1\\ 0\end{array}\right){\vec{e}}_2=\left(\begin{array}{l}0\\ 1\end{array}\right).$ The linear transformation $?:{\mathbb{R}}^2?{\mathbb{R}}^2$ is defined as $?\left(\begin{array}{l}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{r}?x_1+x_2\\ ?3x_1+4x_2\end{array}\right)$ Tasks (a) What is the matrix representation of $?$ from the standard basis to the standard basis? ${\mathcal{R}}_{E?E}(?)=()$ (b) Suppose basis $B=\left\{{\vec{b}}_1,{\vec{b}}_2\right\}$ in ${\mathbb{R}}^2$ is ${\vec{b}}_1=\left(\begin{array}{c}1\\ ?1\end{array}\right){\vec{b}}_2=\left(\begin{array}{c}2\\ ?1\end{array}\right)\text{, }$ what is the matrix representation of $?$ from basis $B$ to basis $B$ ? ${\mathcal{R}}_{B?B}(?)=($ Hint: ${\mathcal{R}}_{B?E}(id)$ is trivial to find. (c) Find a basis $G=\left\{{\vec{g}}_1,{\vec{g}}_2\right\}$ such that the matrix representation of $?$ from basis $G$ to basis $G$ is $\begin{array}{c}{\mathcal{R}}_{G?G}(?)=\left(\begin{array}{cc}39& 23\\ ?61& ?36\end{array}\right)\\ {\vec{g}}_1=(){\vec{g}}_2=\left(\begin{array}{l}\end{array}\right)\end{array}$