I've been learning about tensor analysis, and things have been going well so far, but I'm a bit stuck when it comes to the idea of a cobasis (by which I mean the reciprocal basis; not sure which term is more common). Basis vectors are simple enough, it's just a carry over from vector spaces in linear algebra. Cobasis vectors, on the other hand, don't seem well motivated to me. I understand that they're dual to the basis vectors, but why exactly do we need them? When should they be used over the normal basis vectors?
I'm familiar with the treatment of vectors as elements of a tangent space of a manifold and dual vectors as linear operators on vectors, if that's the sort of explanation that motivates the cobasis most effectively, although I don't have a good intuition with that sort of thing quite yet.