I'm reading Hoffman and Kunze's linear algebra book and on page 73 in the exercise 7, they ask to verify this function
$$T(x_1,x_2,x_3)=(x_1-x_2+2x_3,2x_1+x_2,-x_1-2x_2+2x_3)$$ is a linear transformation.
This exercise is really simple, but a little bit tedious. We have to define arbitrary $u=(x_u,y_u,z_u)$ and $v=(x_v,y_v,z_v)$ elements of $F^3$ and show $T(u+v)=T(u)+T(v)$ and $T(ku)=kT(u)$ for $k\in F$. (we can see $F$ as $\mathbb R$ or $\mathbb C$)
Is there a way more elegant to prove this function is linear?