For vector space $V$ and $v \in V$, there is a natural identification $T_vV \cong V$ For vector space $V$ and $v \in V$, there is a natural identification $T_vV \cong V$ where $T_vV$ is the tangent space of $V$ at $v$. Can somebody explain this identification to me? Thank you
 A: If you had somebody going with a bicycle through any point $v\in V$ then unless he stopped in that point and turned somehow, we can determine the exact direction he went through that point $v$. Now the question is: What means direciton? And the Answer is we take all the derivatives of the way the guy with the bicycle could go through that point (smooth curves) and identify two ways of going through that point if the derivative in that point is multiple of the other.
Now for every $w \in V$ the guy can take a path so that the derivative is just $w$. The other way around is even more obvious. Since we can discribe the path the guy takes with a smooth curve from $\mathbb R \to V$ we get that the derivative in $u$ is in $V$ and therefore the direction is in $T_uV$ which is the set of all directions in $u$.
As a little hint. You can also define $T_uV$ as the derivatives in the point $u$. Anyway both of the definitions are isomorph.
A: $T_vV$ consists of the derivations $\mathcal E_v\to\mathbb R$ on the ring $\mathcal E_v$ of germs of differentiable functions in $v$. Now just identify each vector $x\in V$ with the directional derivative along $x$ evaluated at $v$: $[f]\mapsto\mathrm D_x f(v)$, which is a derivation ($[f]$ is the germ of $f$ in $v$). Essentially, the elements of $T_vV$ are all the directional derivatives evaluated at $v$, and every directional derivative corresponds to a direction (a vector) in $V$, making $T_vV$ and $V$ isomorphic.
A: It is straightforward to check that
$$\Phi_v \colon V \to T_vV, \qquad \Phi_v(x) := \frac{d}{dt}\bigg\vert_{t = 0}(v + tx)$$
is an isomorphism. Note that the definition of $\Phi_v$ makes explicit use of the fact that $V$ is a vector space but does not depend on a choice of basis. Hence $\Phi_v$ provides a canonical identification of $T_v V$ with $V$.
