Vector Spaces versus Affine Spaces

Question

Why did the notion of a vector space become primarly used rather than the notion of an affine space? Are there examples of vector spaces that cannot be viewed as an affine space?

Answer 1

Every vector space can be trivially seen as affine space. Given a vector space $V$, an affine space modelled on $V$ can be defined as a triple $(A,V,\delta)$, where $A$ is a set whose elements are called points and $$\delta\colon A \times A \to V$$ is a function such that

1. For any $P \in A$ and for any vector $v \in V$ there is a unique $Q$ such that $\delta(P,Q) = v$.
2. If $P,Q,R \in A$, then $\delta(P,Q) = \delta(P,R)+\delta(R,Q)$.

Notice that axiom 1 is saying that for each $P \in A$ the function $\delta(P,\cdot)$ is a bijective function, so $A$ and $V$ are in one-to-one correspondence. Axiom 2 is the usual parallelogram law. By abuse of notation, instead of saying that the triple $(A,V,\delta)$ is an affine space, one usually says that $A$ is affine. So $A$ behaves geometrically as $V$, but unlike in vector spaces there is no preferred origin.

Given that, consider the triple $(V,V,\delta)$, where $\delta \colon V \times V \to V$ is simply $\delta(u,v) = u-v$. Then for any $u \in V$ and any vector $v\in V$ there is a unique $r \in V$ such that $\delta(u,r) = u-r=v$, and it is trivially $r = u-v$. Further, for any $a,b,c \in V$ you have $$\delta(a,b)+\delta(b,c) = a-b+b-c=a-c=\delta(a,c),$$ so axioms 1. and 2. are satisfied, and $(V,V,\delta)$ is an affine space.

I am not really sure about why vector spaces became 'primarily used', I cannot say this is true. A comment in this regard might be that vector spaces have generally more structure than (strictly) affine spaces, and crop up frequently. For instance, in differential geometry tangent spaces to differentiable manifolds, tensor algebras, Lie algebras, etc., they all have the structure of vector space in common. A first example of affine space in differential geometry is the set of derivations on a manifold, which is modelled on the vector space of tensor fields of type $(1,1)$. So I would say this kind of structure seems to crop up less frequently, but this is only a comment based on my little experience.