Given a manifold M, I imagine the tangent space at p, $T_p (M)$ as follows:

- Consider a higher dimensional vector H space where M is embedded.
- Look at all smooth curves in M that pass through p.
- Compute the velocity vector of the curve at p. These vectors will lie in higher dimensional vector space H.
- Together these vectors form the tangent space.

The formal definition however is as follows:

For any given smooth curve $lambda$ in M passing through p, define $v_{p, lambda}$ as a linear map that maps smooth functionals on M to a real number as follows:

$$v_{p, lambda}(f) = (fcirc lambda)'(0)$$

where $lambda(0) = p$.

The set of all such linear maps form the tangent space.

How do I reconcile the intuitive picture with the actual definition?