Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-vector space, and restrict scalars along $F$. We get a new $k$-vector space $F_*(k)$, and I wonder how does it look (possibly even when $k$ is perfect)? Can we give a basis for $F_*(k)$? Is it finite dimensional?
I will also be interested if we can say something more generally about $F_*(V)$ for $V$ a finite dimensional $k$ vector space. Thank you
EDIT: I am still interesting in the question. From the comments, we have the following simplification by reuns:
Find a basis for $k$ as a $k^p$-vector space.
I did not understand the rest of reuns' suggestion for how to do it though.