$\operatorname{Hom}_k(k,V)$ is a vector space? Is it true that a vector space is just the set of maps from the underlying field to the space itself.  I.e. if $V$ is a vector space of the field $k$ then
$$
V\cong \operatorname{Hom}_k(k,V)
$$ if so then this would make an intuitive understanding of the dual space $V^*$ somewhat trivial since
$$
V^{**}\cong V\cong \operatorname{Hom}_k(k,V)\implies V^*=\operatorname{Hom}_k(V,k) 
$$
If true, an explanation of why $V\cong \operatorname{Hom}_k(k,V)$ with a simple example or two would provide a lot of clarity for me since I could easily grasp the dual vector space idea from that point.
Edit:  I actually had to read two proposed answers a couple times for the idea to sink in but I could only pick one answer.

Indeed, we have $\DeclareMathOperator{\Hom}{Hom} V \cong \Hom_k(k,V)$.  To see that this is the case, note that we have the (natural) isomorphism $\Phi: V \to \Hom_k(k,V)$ given by
$$
[\Phi(v)](t) = t \cdot v
$$
for $v \in V$ and $t \in k$.
In terms of matrices, this amounts to saying that a column-vector $v = (v_1,\dots,v_n) \in k^n$ can be naturally identified with the $n \times 1$ matrix
$$
\pmatrix{v_1\\ \vdots \\ v_n}
$$
which denotes a linear map from $k^1$ to $k^n$.
