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This is from Frankel's The Geometry of Physics:

Problem 2.3(2) Consider the tangent bundle to a manifold $M$.

  1. Show that under a change of coordinates in $M$, $\partial/\partial q$ depends on both $\partial/\partial q'$ and $\partial/\partial \dot{q}'$.

I'm not entirely sure what $\partial/\partial \dot{q}$ means. If $q = (q^1, \dots, q^n)$ is a set of coordinates for $M$, then $\partial/\partial q^i$ are the basis vectors for the tangent space of $M$ at a given point. The tangent bundle to $M$, $TM$, is a $2n$-dimensional manifold, and so to label a point in $TM$ we need a $2n$-tuple of numbers $(q^1, \dots , q^n, \dot{q}^1, \dots , \dot{q}^n)$. I guess that then $\{\partial/ \partial q^i, \partial/ \partial \dot{q}^i\}$ would be a basis for the tangent space to $TM$.

Is this the correct interpretation? If so, does this imply that the tangent space to $TM$ can be written as the direct sum of two subspaces, the tangent space to $M$ and the tangent space to the tangent space to $M$? Or is this a coordinate dependent identification?

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Your interpretation is correct - given a set of coordinates $q$ on $M$, a tangent vector in the domain of these coordinates can be completely described by its base point $q$ and its component in each of the $\partial/\partial q^i$ directions; so calling the component of the vector in the $\partial/\partial q^i$ direction $\dot q^i$ we have a coordinate system $(q,\dot q)$ for $TM$.

The direct sum decomposition you're talking about indeed depends on coordinates - the "horizontal directions" (i.e. the vectors in $T_p TM$ we would like to identify with $T_p M$) are those in which the components of vectors remain constant while their base points change - this dependence on components is a dependence on the coordinate system. Thus there is no canonical horizontal subspace without some extra structure, which is known as a connection.

The vertical subspace $V_p$ (directions in which the base point is fixed), however, is coordinate independent; thus you can write down a decomposition of the form $T_p TM = V_p \oplus (T_p TM / V_p)$. A connection is then exactly the extra information that identifies the quotient $T_p TM / V_p$ with a subspace of $T_p TM$.

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