Let $a, b \in \mathbb{R}$, and consider the $\mathbb{R}$-linear map $f : \mathbb{C} \longrightarrow \mathbb{C},\ z \mapsto az + b\bar{z}$. Choose the correct statement(s) from below:\\
(A) $f$ is onto (i.e., surjective) if $ab \neq 0$;\\
(B) $f$ is one-one (i.e., injective) if $ab \neq 0$;\\
(C) $f$ is onto if $a^2 \neq b^2$;\\
(D) if $a^2 = b^2$, $f$ is not one-one.