Answer
$T$ is a linear transformation.
Work Step by Step
A linear transformation is said to be linear if
$T(av+w)=aT(v)+T(w)$, for all $v,w$ in the given domain and for all scalars $a$.
Since the domain is $R^3$, take $v=(x_1,y_1,z_1)$ and let $w=(x_2,y_2,z_2)$ be arbitrary elements in $R^3$ and $a$ be any scalar.
$T(a(x_1,y_1,z_1)+(x_2,y_2,z_2))=T((ax_1,ay_1,az_1)+(x_2,y_2,z_2))$
$T(ax_1+x_2,ay_1+y_2,az_1+z_2))$
$=((ax_1+x_2)+(ay_1+y_2),(ax_1+x_2)-(ay_1+y_2),az_1+z_2)$
$=(ax_1+ay_1,ax_1-ay_1,az_1)+(x_2+y_2,x_2-y_2,z_2)$
$=a(x_1+y_1,x_1-y_1,z_1)+(x_2+y_2,x_2-y_2,z_2)$
$=aT(x_1,y_1,z_1)+T(x_2,y_2,z_2)$
$=aT(v)+T(w)$
Thus, $v$ and $w$ are arbitrary.
$T(av+w)=aT(v)+T(w)$, for all $v,w$ in $R^3$ and for all scalars $a$.
Hence, $T$ is a linear transformation.
