Answer
See solution below
Work Step by Step
a) $T(\vec{e_1})=-\vec{b_2}+\vec{b_3}$
$T(\vec{e_2})=-\vec{b_1}-\vec{b_3}$
$T(\vec{e_3})=\vec{b_1}-\vec{b_2}$
b) $[T(\vec{e_1})]_B=\begin{bmatrix}
0\\
-1\\
1
\end{bmatrix}$, $[T(\vec{e_2})]_B=\begin{bmatrix}
-1\\
0\\
-1
\end{bmatrix}$, and
$[T(\vec{e_3})]_B=\begin{bmatrix}
1\\
-1\\
0
\end{bmatrix}$
c) The matrix T relative to $E$ and $B$ is
$\begin{bmatrix}
0&-1&1\\
-1&0&-1\\
1&-1&0
\end{bmatrix}$