An example of the "natural" paring $V^* \times V \rightarrow \mathbb{R}$ This so called natural paring is not natural to me at all. I am wonder if someone could give me an explicit example? I understand that $V^*$ is the dual space of $V$, and to my understanding, its basis is $f^*$, which are row vectors looks like $(1, 0, \dots, 0)$. Thank you.

By definition, if $V$ is a vector space (forget about it having a basis) over a field $k$, then the dual space $V^*$ is the space of linear functionals $f\colon V\to k$. Thus the most natural thing to do if you have a linear functional $f\in V^*$ and a vector $v\in V$, is to take $f(v)$. This gives you the natural pairing $V^*\times V\to k$, $(f,v)\mapsto f(v)$.