I am trying to solve the following problem. I am stuck while I was thinking of the tangent vectors.
How do I construct an explicit diffeomorphism between $TS^1$ and $S^1\times\Bbb{R}$?
So my question is how does $TS^1$ look like? I know it is disjoint union of Tangent Spaces at every point $z \in S^1$. But can anyone explain how does it look like in $R^2$?
I think it is the tangent at the point $z\in S_1$. I cannot see how that is a vector space. I read that the tangent space is a vector space too. Can anyone explain me with a diagram. It would help a lot.
Edit 1: Wikipedia shows the red coloured lines as tangent space of a point. But isn't it supposed to be a vector space?
https://en.m.wikipedia.org/wiki/Tangent_bundle#cite_note-disjoint-1