Prove that if $\phi$ is injective then the dual function $\phi^{*}$ is surjective

Question

V,W finite dimensional K-vector spaces. $\phi:V \rightarrow W$ linear map

I have proven that if $\phi$ is surjective then $\phi^{*}$ is injective. Now I have to prove that if $\phi$ is injective then $\phi^{*}$ is surjective and I don't have a clue how. I tried to work with dimensions. I tried to prove that dim(Im($\phi^{*}$)) is greater or equal than dim(($W^{*}$)) but was unable to.

I've also tried to take g belonging to $V^{*}$ and find a f belonging to $W^{*}$ such that $\phi^{*}(f)$=g but was unable to

Answer 1

So I first misread your question, but this should actually be correct now. Given that $\phi : V \to W$ is injective, we have $V \cong \phi(V) \subseteq W$. Let $\psi : \phi(V) \to V$ be its inverse. Now given any $f: V \to k$, we can consider $\tilde f = f \circ \psi : \phi(V) \to k$ and extend it linearly to a map $\hat f : W \to k$. Now consider $\phi^*(\hat f)$. This is just $f$ and therefore $\phi^*$ is surjective.