TITLE: Why do we care about dual spaces?
QUESTION [238 upvotes]: When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra.
I was wondering if anyone knew a nice intuitive motivation for the study of dual spaces and whether or not they "show up" as often as other concepts in linear algebra? Is their usefulness something that just becomes more apparent as you learn more math and see them arise in different settings?

Edit
I understand that dual spaces show up in functional analysis and multilinear algebra, but I still don't really understand the intuition/motivation behind their definition in the standard topics covered in a linear algebra course. (Hopefully, this clarifies my question)

REPLY [4 votes]: To take the case in $ \mathbb{R}^n $
If $ A $ are the Transformation Matrix from the Natural Basis to an Arbitrary Standard Basis then $ A^{-1} $ are the Map from the Natural Basis to the Dual Basis. That $ A^{-1} $ exist is guaranteed as long as the Standard Basis Span the same Vector Space as the Natural Basis and are Linear Independent (which it by definition should be).
Dual Basis <---> Natural Basis <---> Standard Basis
Let $ \vec{e}_{\alpha} = \vec{e}^{\ \alpha} $ be Natural Basis
Let $ \vec{e}_{\widetilde{\alpha}} $ be the Standard Basis
Let $ \vec{e}^{\ \widetilde{\alpha}} $ be the Dual Basis
$ \vec{e}_{\widetilde{\alpha}} = \sum_{\alpha} A^{\alpha}_{\ \ \widetilde{\alpha}} \ \vec{e}_{\alpha} $
$ \vec{e}_{\widetilde{\alpha}} = \sum_{\alpha} \sum_{\widetilde{\beta}} A^{\alpha}_{\ \ \widetilde{\alpha}} \ A^{\alpha}_{\ \ \widetilde{\beta}} \vec{e}^{\ \widetilde{\beta}} $
Note that the two A's in the middle can be substituted with the Metric Tensor.
$ \vec{e}_{\widetilde{\alpha}} = \sum_{\widetilde{\beta}} g_{\widetilde{\alpha}\widetilde{\beta}} \vec{e}^{\ \widetilde{\beta}} $
The "Engineer's Approach" to Dual Space would be:
1) Some Vector Components $ (x^{\alpha}) $ have the unit $[m]$
2) Some Vector Components $ (x_{\alpha}) $ have the unit $[1/m] \ $ (e.g. gradient)
3) When Multiplying Components with Bases the resulting Vector live in $ \mathbb{R}^n $ and have the unit $ [1] $ 
4) Therefore; Need two different Bases