I am trying to get familiar with the idea of a vector bundle at an intuitive level, and it doesn't seem to be too abstract an idea: vector spaces get "attached" to points on a surface.
However, there is a naive, and early question that comes to mind reading the definition on Wikipedia:
... to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X
It is one of the axioms of a vector space that there is an identity element of addition:
There exists an element $0 ∈ V,$ called the zero vector, such that $v + 0 = v$ for all v ∈ V
which I interpret to mean that vector spaces are centered at the origin of the coordinate system.
So in the case of a vector space attached to a point on some surface, is the zero the actual point?