Eigenvalues of invariant subspaces

Question

Let $V$ be a vector space over $F$ and $f:V\mapsto V$ a linear map. If $dimV=n \geq 2$ and the only invariant subspaces of $V$ are $V$ itself and {$0_V$} ,then investigate if $f$ has eigenvalues.

I'm sorry if this is a trivial question, but my book has a little to no info about invariant subspaces or even the definition of what invariant really means and what follows that.

Any little i know about it comes from wikipedia so if anyone has some site or thread(question on this site) to propose it would be great.

Answer 1

If $f$ has some eigenvalue $\lambda$, let $v$ in $V\setminus\{0\}$ be such that $f(v)=\lambda v$. Then, for each $\mu\in F$,$$f(\mu v)=\mu f(v)=\,u\lambda v=\lambda(\mu v).$$So, $Fv$ is a one dimensional invariant subspace of $V$. But it is being assumed that $V$ has invariant subspaces other than $V$ or $\{0_V\}$, and $Fv$ is none of them.