Let $V, W$ be finite-dimensional vector spaces over a field $F$ and $f: V \rightarrow W$ a linear map.

(i) Show that $f$ is injective if and only if the image of every linearly independent subset of $V$ is linearly independent in $W$.

(ii) Define the dual space $V^{*}$ of $V$ and the dual map $f^{*}: W^{*} \rightarrow V^{*}$.

(iii) Show that $f$ is surjective if and only if the image under $f^{*}$ of every linearly independent subset of $W^{*}$ is linearly independent in $V^{*}$.