(1 point) Let ${\overrightarrow{b}}_1=\left[\begin{array}{c}-1\\ 3\end{array}\right]$ and ${\overrightarrow{b}}_2=\left[\begin{array}{c}3\\ -8\end{array}\right]$. The set $B=\{{\overrightarrow{b}}_1,{\overrightarrow{b}}_2\}$ is a basis for ${\mathbb{R}}^2$. Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a linear transformation such that $T\left({\overrightarrow{b}}_1\right)=2{\overrightarrow{b}}_1+6{\overrightarrow{b}}_2$ and $T\left({\overrightarrow{b}}_2\right)=2{\overrightarrow{b}}_1+2{\overrightarrow{b}}_2$.
(a) The matrix of $T$ relative to the basis $B$ is
(b) The matrix of $T$ relative to the standard basis $E$ for ${\mathbb{R}}^2$ is

Step 1 ：Given the basis vectors ${\overrightarrow{b}}_1=\left[\begin{array}{c}-1\\ 3\end{array}\right]$ and ${\overrightarrow{b}}_2=\left[\begin{array}{c}3\\ -8\end{array}\right]$ for ${\mathbb{R}}^2$, and the linear transformation T such that $T({\overrightarrow{b}}_1)=2{\overrightarrow{b}}_1+6{\overrightarrow{b}}_2$ and $T({\overrightarrow{b}}_2)=2{\overrightarrow{b}}_1+2{\overrightarrow{b}}_2$.

Step 2 ：To find the matrix of T relative to the basis B, we express the images of the basis vectors under T as linear combinations of the basis vectors. The coefficients of these linear combinations form the entries of the matrix. For $T({\overrightarrow{b}}_1)$, the coefficients are 2 and 6, and for $T({\overrightarrow{b}}_2)$, the coefficients are 2 and 2. Therefore, the matrix of T relative to the basis B is $\boxed{{\displaystyle \left[\begin{array}{cc}2& 2\\ 6& 2\end{array}\right]}}$.

Step 3 ：To find the matrix of T relative to the standard basis E for ${\mathbb{R}}^2$, we express the images of the standard basis vectors under T as linear combinations of the standard basis vectors. The coefficients of these linear combinations form the entries of the matrix. For $T({\overrightarrow{e}}_1)$, the coefficients are 16 and -42, and for $T({\overrightarrow{e}}_2)$, the coefficients are 4 and -10. Therefore, the matrix of T relative to the standard basis E for ${\mathbb{R}}^2$ is $\boxed{{\displaystyle \left[\begin{array}{cc}16& 4\\ -42& -10\end{array}\right]}}$.