Exercises
Ex. 01
Assume $\varphi$ is not the zero map. Then for some vector $v_0$, $\varphi(v_0) = \lambda_0 \neq 0$. Let $\lambda \in F$ be arbitrary. Then by linearity $\varphi(\lambda \cdot (\lambda_0)^{-1} v_0) = \lambda (\lambda_0)^{-1} \lambda_0 = \lambda$. Hence surjective.