Is $\textit{affine space}$ the same as $\textit{quotient space}$?

Question

From the answer for this question, I understand that affine subspace is the same as affine subset, however (despite the somewhat misleading question's title), it doesn't say that affine space is the same as quotient space.

Also, I found the following definition of affine space on Wikipedia:

An affine space is a set A together with a vector space $\overrightarrow{A}$ and a transitive and free action of the additive group of $\overrightarrow{A}$ on the set $A$.

Which uses some terminologies, probably in Group theory, that I'm not familiar with. Some other "intuitive" explanations suggest to me that affine space is actually the same as quotient space.

Is that true that affine space is a synonym of quotient space?

Answer 1

The vocabulary "transitive and free action" refers to group action on a set.

You know vectors and points and relation between three points $A,B, C$:$$\vec{AB}+\vec{BC}=\vec{AC}$$ Describing an affine space as in wikipedia definition you refer in OP is an equivalent way of describing a point space where two points $A$ and $B$ define a vector $\vec{AB}$.

Let $M$ a point and $u$ a vector. $$M+u \text{ refers to the point }M' \text{ s.t. }u=\vec{MM'}$$ In accordance with my comment, i.e. the translation of $M$ by the translation $t_u$ of vector $u$.

This equivalent definition in terms of the action of the additive group of vectors and this notation "$M+u$" is very convenient to define an affine subspace of $(A,\vec A,+)$ as $$M+V$$where $M\in A$ and $V$ is a linear subspace of $\vec A$.

$$\text{V is called the direction of M+V}$$

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Now, if you consider the equivalence relation $\mathcal R$ on $A$ defined by$$M\mathcal R N\iff N\in M+V$$the equivalence classes are precisely the affine subspace of direction $V$.

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Answer 2

The concepts of "affine space" and "quotient space" are by no means identical. In the linked question it was observed that the elements of a quotient space $V/V_0$, where $V$ is a vector space and $V_0 \subset V$ is a linear subspace, are the subsets of $V$ having the form $x + V_0$ with $x \in V$. Thus $$V/V_0 = \{ x + V_0 \mid x \in V \} .$$

Clearly $x + V_0 = y + V_0$ iff $x - y \in V_0$.

For $x \notin V_0$ the set $x + V_0$ is not a linear subspace of $V$; it is a translated copy of $V_0$. The sets $x + V_0$ are called affine subspaces of $V$.

To understand the concept of an affine subspace you do not need to know the general concept of an affine space.

However, each affine subspace $x + V_0$ (which is no vector space unless $x \in V_0$, which means $x + V_0 = V_0$) can be regarded as affine space in the sense of the Wikipedia-definition. We simply define an action of $V_0$ on $x + V_0$ by $$(u, x + v) \mapsto x + v + u .$$

You can easily verify the properties required in Wikipedia.