Answer
See below
Work Step by Step
Take $T\begin{bmatrix}
1 & 0\\
1 & 1
\end{bmatrix}=(1,0)\\
T\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}=(0,1)$
Then, $T\begin{bmatrix}
1 & 0\\
1 & 1
\end{bmatrix}+T\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}=(1,0)+(0,1)=(1,1)$
But, $T(\begin{bmatrix}
1 & 0\\
1 & 1
\end{bmatrix}+\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix})=T\begin{bmatrix}
2 & 1\\
1 & 2
\end{bmatrix}=(2,2)$
Hence, $T(\begin{bmatrix}
1 & 0\\
1 & 1
\end{bmatrix})+T(\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}) \ne T(\begin{bmatrix}
1 & 0\\
1 & 1
\end{bmatrix}+\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix})$
$T$ is not a linear transformation.
