Every affine spaces are vector spaces
Not every vector spaces are affine spaces.
Why this happens? I read HERE definition of affine space
An affine space is a vector space acting on a set faithfully and transitively
In other word, an affine space is always a vector space but why, in algebraic terms not every vector spaces are affine spaces? Maybe because a vector space can also not acting on a set faithfully and transitively? But in what way can you show me this using group theory/torsor?
But reading on wikipedia definition of affine space it's not a vector space that acts on something, but the reverse: a set that's acted on by a vector space. The vectors acting on the affine space are usually rebranded as “translations”. Technically an affine space is a pair of things: $A$ “together with” $V$, but the set of points is definitely $A$, not $V$.
I also take a look about G-torsor theory because an affine space is a principal homogeneous space.
Every group G can itself be thought of as a left or right G-torsor under the natural action of left or right multiplication.
Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly by saying that $A$ is a principal homogeneous space for $V$ acting as the additive group of translations.
Definition
Let $G$ be a group. The theory of G-torsors $\mathbb{T}_G$ is the geometric theory over the signature with one sort $U$, a set of unary functions symbols ${g\in G}$ and the following axiom scheme:
$$ \begin{aligned} \top & \vdash (g_i g_j)u=g_i(g_j u)\quad\text{for all}\; g_i,g_j \\ \top&\vdash (\exists u)\; \top \quad\text{(U is inhabited)} \\ g_{i}u = g_{j}u &\vdash \bot\quad \text{for all pairs}\;g_i\neq g_j\quad\text{(G acts freely)} \\ \top &\vdash (\exists x) \bigvee_{g\in G} gx=y\quad \text{(G acts transitively)}\quad . \end{aligned} $$
