Answer
$A=\begin{bmatrix}
1 & -1 & 1\\
-1 & 0 & 0
\end{bmatrix}$
Work Step by Step
We are given:
$T(x_1,x_2,x_3)=(x_1-x_2+x_3,x_3-x_1)$
The standard basis vectors in $R_3$ are:
$e_1=(1,0,0)\\
e_2=(0,1,0) \\
e_3=(0,0,1)$
Consequently,
$T(e_1)=(1,-1)\\
T(e_2)=(-1,0)\\
T(e_3)=(1,1)$
The matrix of the given transformation is:
$A=T(e_1,e_2,e_3)=\begin{bmatrix}
1 & -1 & 1\\
-1 & 0 & 0
\end{bmatrix}$
