Let $V$ be a finite-dimensional vector space over a field $k$. In a calculation I found myself wondering about this:
Is picking a monomorphism (subobject) $S\to V$ equivalent to picking an element of the dual vector space $V^*$ of $V$?
I can not convince myself on whether this is true or not, so I would appreciate any help.
This doesn't seem to work particularly well. The closest thing that makes sense to me here is to associate to an element of $V^*$ -- that is, a mapping $V \to k$ -- its kernel, which is indeed a subobject (subspace) of $V$. However, it generally covers only few of the subobjects, namely those with codimension 1. For example, in $\mathbb R^3$, no element of $V^*$ corresponds to the inclusion $\mathbb R \to \mathbb R^3: x \mapsto (x,0,0)$. Also, it covers some of the subobjects multiple times; namely, if $f \in V^*$, then any non-zero scalar multiple of $f$ gets associated to the same subspace.