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.

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    One of the (numerous) things that make the dual space a very useful tool is the correspondance between hyperplanes in $V$ and lines in $V^*$. It gives another point of view on a lot of reduction question. – TheSilverDoe Dec 26 '21 at 16:33
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    @TheSilverDoe can you please elaborate more, or point me to some material? I'm a 5th semester pure math undergraduate – AyamGorengPedes Dec 26 '21 at 16:35
  • Duality is a deep mathematical concept in general, not only here for vector spaces. – Dietrich Burde Dec 26 '21 at 17:45
  • I think the usual approach to dual spaces in most treatments of finite-dimensional linear algebra obscures the big picture. You might find my video helpful: https://youtu.be/eOIJzb7SItg – blargoner Dec 26 '21 at 18:35
  • @DietrichBurde I can't help but think so, I have encountered duals several times, such as the dual of a linear program (we went through the geometry for the primal, not for the dual), or topological and algebraic duals of a graph. Each time it just exists, and we use it, but I don't understand a deeper meaning of it. – AyamGorengPedes Dec 26 '21 at 18:43

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