Consider the transformation $T$ from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$ given by
\[
T\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]=x_{1}\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right]+x_{2}\left[\begin{array}{l}
4 \\
5 \\
6
\end{array}\right]
\]
Is this transformation linear? If so, find its matrix.