Answer
$\left[ {\begin{array}{*{20}{c}}
2&{ - 3}\\
1&{ - 1}\\
{ - 4}&1
\end{array}} \right]$
Work Step by Step
Take the standard basis for $R^2$, which is $(1,0),(0,1)={e_1,e_2}$.
Then, $T(1,0)=(2(1)-3(0),1−0,0-4(1))=(2,1,-4)\\
T(0,1)=(2(0)-3(1),0-1,1−4(0))=(-3,−1,1)$
Therefore, the matrix corresponding to the linear transformation $T$ is given by:
$A=[T(e_1) T(e_2)]\\
=\left[ {\begin{array}{*{20}{c}}
2&{ - 3}\\
1&{ - 1}\\
{ - 4}&1
\end{array}} \right]$
