I write $P + v$ for the action of the vector $v$ on the point $P$.
An affine space is, in fact, probably the very first visualization you had of vector spaces: e.g. you thought of the plane full of points, and that vectors were arrows that went from one point to another.
The concept of affine space I know requires the action of $V$ on $X$ to be transitive and faithful: this means that, in an affine space, we can define subtraction: $P - Q$ is the unique vector $v$ such that $Q + v = P$. The pair $(Q, v)$ can be pictured as an arrow from $Q$ to $P$.
We can even define nearly arbitrary linear combinations of points: the restriction is that the coefficients have to sum to zero (thus giving us a vector) or sum to one (thus giving us a point).
e.g. if I write $P + \frac{1}{2}Q - \frac{3}{2} R$, I 'really' mean the vector
$$P + \frac{1}{2}Q - \frac{3}{2} R = (P - R) + \frac{1}{2} (Q - R)$$
or any other similar rearrangement into "legal" operations. (they would all give the same answer)
Similarly, if I write $\frac{1}{2}P + \frac{1}{3} Q + \frac{1}{6} R$, I mean the point
$$ P + \frac{1}{3} \left(Q - P \right) + \frac{1}{6} \left(R - P \right) $$
I believe I've seen some definitions of affine space that don't make reference to vectors at all: they are instead axiomatized in terms of the arithmetic of points.
The notion of vector space is, in fact, equivalent to the notion of a pair consisting of an affine space and a point on the space. A vector space already has the structure of an affine space; it just comes equipped with a distinguished point $0$. Conversely, given any affine space and a choice of a point $O$, we can complete its vector space structure by treating every other point $P$ as the vector $P - O$.
Beside from thinking like vectors, convex combinations (where the coefficients sum is one and are all positive) can be thought of as averaging between points. For example,
- $\frac{1}{3} P + \frac{2}{3} Q$ is the point on the line segment $PQ$ that lies two thirds of the way from $P$ to $Q$
They're called convex combinations, because of the relation to convex sets: any convex combination of points in a convex set is again in the set. In fact, the convex hull of a set of points is precisely the set of all convex combinations of those points.
e.g. the set of all points inside the triangle $\Delta PQR$ are of the form $aP + bQ + cR$ where $a+b+c = 1$ and $a,b,c$ are all positive. (if we allow $a,b,c$ to be zero as well, then this set of points would include the triangle itself)
More general affine combinations (the coefficients sum to one) are similar:
- $2P - Q$ is the point on the line $PQ$ that lies on the other side of $P$ from $Q$ that is twice as far from $Q$ as it is from $P$.
All convex combinations, actually, can be made out of binary ones: for example,
$$ \frac{1}{2} P + \frac{1}{3} Q + \frac{1}{6} R = \frac{5}{6} \left(\frac{3}{5} P + \frac{2}{5} Q \right) + \frac{1}{6} R $$
(how did I find this? Let $S$ be this point. I picked points on the plane for $P$, $Q$, and $R$ and found where the line $RS$ met the segment $PQ$: that point is the point that appears in the parentheses on the right hand side)
For linear combinations, you can just pretend the points are vectors. It doesn't matter which point you choose in the affine plane as the origin, you'd get the same result. e.g. my earlier example satisfies
$$P + \frac{1}{2}Q - \frac{3}{2} R = (P - O) + \frac{1}{2} (Q-O) - \frac{3}{2} (R - O)$$
no matter which point you choose for $O$.
Best Answer
For a linear space $V$, the corresponding affine space is $(V, V, \phi)$ where $\phi(v, w) =w-v$ (or $v+w$, depending on your definition) is the canonical action.