Linear Algebra Question from Mock GRE exam. Suppose $V$ and $W$ are finite dimensional vector spaces, and that $f:V\mapsto W$ is a linear map. Suppose $\{e_1,...,e_n\}\subset V$ and that $\{f(e_1),...,f(e_n)\}$ is a basis of $W$. Then which of the following are true?

I. $\{e_1,...,e_n\}$ is a basis of $V$

II. There exists a linear map $g:W\mapsto V$ such that $g\circ f=Id_V$

III. There exists a linear map $g:W\mapsto V$ such that $f\circ g=Id_W$

(A) I only

(B) II only

(C) III only

(D) I and III

(E) II and III



The solutions manual says it is (B), but I think it should be (C) since we can regard $W$ to be isomorphic to some subspace of $V$? 

Thanks in advance!
Consider III. Suppose it is true.
 $$w \in W \implies w=fg(w)=f(g(w))\\\implies w \in \;\;\text{range}\; f$$ Hence $ \text{range}\; f=W$, concluding $f$ is onto.

Is $f$ onto ?
For sure the I. it's false infact V could be $\mathbb{R^{n+1}}$ and $f(e_{n+1})=0$ and the hypothesis still true. The II. could be false infact if as in the first example $f(e_{n+1})=0$ we have that $g(f(e_{n+1}))=g(0)=0$ so the rank of $g(f(v))$ is $n$ and not $n+1$. So the right one is the III. infact if you choose $g$ such that $g(f(e_n))=e_n$ we have that $f(g(v))$ on $W$ in the base ${f(e_1),f(e_2),...,f(e_n)}$ is the identity.

If the answer it's wrong i will fix it tomorrow.