Tangent spaces of affine space

Question

An affine space of dimension n on $\mathbb R$ is defined to be a non-empty set $E$ such that there exists a vector space $V$ of dimension n on $\mathbb R$ and a mapping

$\phi:E \times E \rightarrow V,\space\space\space (A,B) \mapsto \phi(A,B):=\vec {AB}$

that obeys the following properties:

(i) For any point $O \in E$, the function

$\phi_O: E \rightarrow V,\space\space\space M \mapsto \vec {OM}$

is bijective.

(ii) For any triplet $(A,B,C)$ of elements of $E$, the following relation holds:

$\vec {AB} + \vec {BC} = \vec {AC}.$

People always say that the tangent space at a point $p$ is $V$ but I am having trouble in proof that. How can we proof that tangent space is $V$? My definition of tangent vector $X$ is the linear map $X(f)=\frac{d}{dt}f(\gamma(t))$ where $f$ is real valued function on a manifold and $\gamma$ a curve on a manifold

Answer 1

More precisely, for any $A \in E$ there is a natural isomorphism $T_A E \stackrel{\cong}{\to} V$.

Hint

1. By axiom (i) a choice of point $O \in E$ defines a bijection $\phi_O := \phi(O, \,\cdot\,): E \to V$, and we can declare this bijection to be a diffeomorphism. Axiom (ii) then implies that the smooth manifold structure defined on $E$ this way does not depend on the choice of $O$.
2. Recall that for any $v \in V$ there is a canonical isomorphism $\Psi_v : T_v V \stackrel{\cong}{\to} V$,