linearity and nonlinearity of the PDEs below I just want to verify and check my understanding to see if I can do this right.
a)$ u_x + u_y = 0$
This is linear since: $(u+v)_x + (u+v)_y = u_x +u_y + v_x+v_y $
and $c(u_x+u_y) = cu_x + cu_y$
b) $u_x + yu_y = 0 $
This is linear since:
$(u+v)_x + y(u+v)_y = u_x+v_x + yu_y + yv_y $
and $c(u_x+yu_y) = cu_x+ ycu_y$
c) $u_x + uu_y = 0$
Linear again, I don't see the fail of linearity on this example as well.

The equation in c) is non-linear. For example, take any $\alpha \in \mathbb{R}$ and define the operator $$Lu = u_x + u u_y.$$
Now compute
$$
L(\alpha u) = \alpha u_x + \alpha^2 u u_y,
$$
which shows that $L(\alpha u)$ is in general different than $\alpha Lu$.
