Answer
$\left[ {\begin{array}{*{20}{c}} 1&1&0\\ 1&{ - 1}&0\\{-1}&0&1 \end{array}} \right]$
Work Step by Step
Take the standard basis for $R^3$, which is
${(1,0,0),(0,1,0),(0,0,1)}=\{{e_1,e_2,e_3}\}$.
Then,
$T(1,0,0)=(1+0,1-0,0-1)=(1,1,-1)\\
T(0,1,0)=(0+1,0-1,0-0)=(1,-1,0)\\
T(0,0,1)=(0+0,0-0,1-0)=(0,0,1)
$
Therefore, the matrix corresponding to the linear transformation $T$ is given by:
$A=[T(e_1)$ $T(e_2)$ $T(e_3)$] $=\left[ {\begin{array}{*{20}{c}} 1&1&0\\ 1&{ - 1}&0\\{-1}&0&1 \end{array}} \right]$
