For two vectors $(x_1,y_1,u_1,v_1)$ and $h(x_1,y_1,u_1,v_1)$,
$h(x_1,y_1,u_1,v_1)+h(x_1,y_1,u_1,v_1)$ $=(2x_1 + 3y_1 - u_1 + 2v_1, x_1 - 5y_1 + 6v_1, 2y_1 + u_1 + v_1)+(2x_2 + 3y_2 - u_2 + 2v_2, x_2 - 5y_2 + 6v_2, 2y_2 + u_2 + v_2)$
and add the two vectors component wise, and you get $h(x_1+x_2,y_1+y_2,u_1+u_2,v_1+v_2)$.
Similarly, for a real number $a$, if you compute $a(h(x_1,y_1,u_1,v_1))$, and multiply the scalar and the vector, you will end up with the formula for $h(ax_1,ay_1,au_1,av_1)$.
These are the two things you need to check for a linear transformation.