I can't help but think that I still don't get the big picture about the dual space of a finite vector space. I would like to first show what I understand, and ask for feedback on what I'm lacking or misunderstood. Then, I would also like to know why is the dual space interesting.

Given a finite vector space $V$ over a field $\mathbb{F}$, with basis $\{v_i\}_{i=1}^n$, we have $V^*=\mathcal{L}(V,\mathbb{F})$ the space of all linear functions $f:V\rightarrow\mathbb{F}$, with basis $\{f_i\}_{i=1}^n, f_i(v_j)=\delta_{ij}$. We can then have:

$$ \forall f \in V^*, f = \alpha_1 f_1 + ... + \alpha_n f_n. $$

But I also like to think that:

$$ \forall f\in V^*, f = f(v_1)f_1 + ... + f(v_n)f_n, $$

because given $v \in V, v = a_1v_1 + ... + a_nv_n$,

$$ f(v) = a_1f(v_1)+...+a_nf(v_n)=f(v_1)f_1(v)+...+f(v_n)f_n(v). $$

But I'm not sure in why is this helpful. Also I have also seen the isomorphism $\varphi:V\rightarrow V^*$, where we take $\varphi(v_i)=f_i$, and so we get that $\forall v \in V,v = a_1v_1 + ... + a_nv_n$, we get a corresponding vector in $V^*, f = a_1f_1 + ... + a_nf_n$. So, we can identify an element of $V$ with an element of $V^*$, but I don't understand why is this helpful.

**Edit:** Okay, so I found Andrea Mori's answer, his 3rd point is interesting. I did read about his answer before asking this, but I didn't understand it the first time I read it. I would love to see more answers though, helps my understanding.