Well, no. The first definition doesn't require an ambient space, so can be applied in settings where there is no ambient vector space to host the affine space. Also, relative to your definition, the first definition can put the space in "impossible" places, for example, $i$ units away origin in the host space.

Also, there are non-Euclidean affine spaces (which necessarily cannot be embedded in a (real) vector space). See math.stackexchange.com/questions/2159756/… .

Can you provide an example of a non-euclidean space where my definition fails, with a demonstration of how it fails? The distinction between Euclidean and Minkowskian space depends on a metric. An affine space has no inherent metric.

I'm not understanding what you mean by $i$ units away form the origin in host space. You seem to be talking about something having to do with imaginary numbers. Dr. Moretti specifies that $V$ is a real number vector space.

Your definition fails to distinguish between points and vectors.

A vector space already comes equipped with an origin (the zero vector) and a way to change origin (so to speak), so you seem to be defining an affine space to be a vector space with a possibly different choice of origin $O$. This is backwards: an affine space is supposed to have '$1$ less' origin than a vector space, not '$1$ more.'

An affine space can be defined as an n-dimensional space with coordinates $r^j$ subject to transformations of the form $r^j=\mathrm{e}^j_{\bar{k}}r^{\bar{k}}+\mathrm{r}_{o}^j$ with constant coefficients $\mathrm{e}^j_{\bar{k}},\mathrm{r}_o^j$ where $\det(\mathrm{e}^j_{\bar{k}})\neq 0$. With that definition, $\mathrm{r}_o^j$ serves as an origin.

@Filippo I defined point as distinct from vectors. Points transform under translation as do radial position vectors.

The point of using affine spaces is to avoid defining an origin. While it's often useful to choose one for proofs I think in the definition it should be avoided.

You say that affine space is a set of poiints and that $V$ is a vcector space, but you don't say whether $V$ is the affine space of points. Then you say that given a vector space $V$, we designate $\vec{O}$ to be an origin. But by the definition of a vector space, $V$ already has an origin, so you cannot designate another vector (point?) to be an origin..

@CyclotomicField I didn't ask if my definition is in poor taste. I like Dr. Moretti's definition. The definition in terms of transformation in my comment follows Schouten archive.org/details/isbn_9780486655826/page/n17/mode/2up Which I take as authoritative. Moretti's definition is close to what Weyl gives in Space-Time-Matter. In my view, the most significant aspect of affine geometry is the metrical anarchy. It is often stated that affine space has no metric. That isn't correct. Any set of linearly independent directions can be identified with the standard basis.

A natural model of $n$-dimensional affine space is an $n$-dimensional hyperplane $A \subset V$, where $V$ is an $(n+1)$-dimensional vector space and $A$ does not contain the origin.

@Deane I could have used the zero vector of $V$ as the first choice of origin. I chose arbitrarily. I didn't really think through all the steps of starting with $\mathbf{0}$. I expect it would be a bit more difficult. The reason for arbitrarily designating the origin point of $\mathbb{A}^n$ was to emphasize the anarchy of origins. My question is whether or not my definition is equivalent to Moretti's.

@StevenThomasHatton, what you should do is to show that your definition of affine space satisfies all of the properties in Moretti's definition.

@Deane I probably will do that formally. It's pretty obvious that there is, at least an equivalence. The biggest question is what can "point" mean in Moretti's definition. Whatever a point is, we can add a vector to it and get back another point. So one might say "a point is a thing in physical space" and a vector is just a number and a direction giving instructions to locate another point. So points and vectors are distinct types of things. But that seems disingenuous.

That’s exactly the point of affine space. In Euclidean geometry, you never add two points. It makes no sense geometrically. A vector $v$ is simply a line segment with a direction where you designate which endpoint is the start and which is the end. In Euclidean geometry, given a point $p$, you can put the start of the line segment at $p$ and define $p+v$ to be the end of this line segment. All of this is done without any reference to an origin. And points are definitely not line segments and therefore not vectors.

@Deane Adding two points in Euclidean space makes no sense to me. I didn't define the addition of points. In my definition points aren't invariant under translation, so they can't be added as vectors. Their differences are, however, invariant vectors.

Agreed. Why did you say this was disingenuous?

Saying a point is a physical thing to which we can add a mathematical thing to get another physical thing doesn't seem honest.

I think that you are trying to identify a point $P\in A$ with the function $A\to V,O\mapsto P-O$, i.e. the function that associates to each choice of origin the induced representation of $P$.