Newbie vector spaces question So browsing the tasks our prof gave us to test our skills before the June finals, I've encountered something like this:
"Prove that the kernel and image are subspaces of the space V: $\ker(f) < V, \operatorname{im}(f) < V$, where $<$ means a subspace."
Is it just me or there's something wrong with the problem? I mean: I was rewriting the tasks from the blackboard by hand so I may have made a mistake but is a problem like this solvable or I messed up and should rather look for the task description from someone else? Cause for the time being, I don't see anything to prove here since we don't know what V is, right?

Let $f$ is a linear transformation from $V$ to $V$. So what is the kernel of $f$? Indeed, it is $$\ker(f)=\{v\in V\mid f(v)=0_V\}$$ It is obvious that $\ker(f)\subseteq V$. Now take $a,b\in F$ the field associated to $V$, and let $v,w\in\ker(f)$. We have $$f(av+bw)=f(av)+f(bw)=af(v)+bf(w)= 0+0=0$$ So the subset $\ker(f)$ has the eligibility of being a subspace of $V$. Go the same way for another subset.
