Affine Space Intuition

Question

In a physics text, I was presented with the brief description of an Affine Space. Would it be a correct interpretation to think of $A^n$ as the set of vectors in $\mathbb{R}^n$ that are "floating" or not rooted at the origin? Imagine taking a vector $v\in \mathbb{R}^n$ and translating the arrow that we picture in our mind by $b\in \mathbb{R}^n$. So the $+$ featured in the above description is not the $+$ of the vector space $\mathbb{R}^n$, correct? I'm not too familiar with group algebra but that's what I'm getting from context clues.

Answer 1

Yes, you are correct. Affine space is a vector space with no origin. The $+$ notation used above is not the addition from vector space. Vector space has two operations; addition and scalar multiplication. In this case, we are only considering addition, so that we can deal vector space simply as a group.

Think $+b$ simply as a function. It is the function that changes points of affine space to other point. And the functions form a group(induced from vector space). I think this is the most intuitive explanation of group action.

Answer 2

Yes, that's more or less the intuition.

Since you say it's a physics text, you may prefer a physical intuition. Say the motion of a particle is described by some trajectory $\gamma: \mathbb{R} \to \mathbb{R}^3$. At each time $t \in \mathbb{R}$, we think of the vector $\gamma(t)$ as an arrow pointing from the origin to the position of the particle. The usual way to calculate the velocity of the particle is to just take a derivative of $\gamma$ in the usual way, giving $\dot{\gamma}$. The vector $\dot{\gamma}(t)$ represents the instantaneous direction of travel of the particle at time $t$.

However, the particle is moving in this direction from the point $\gamma(t)$, so that (geometrically) we are concerned with the "vector" $\dot{\gamma}(t)$ emanating from the new origin $\gamma(t)$. This motivates the desire for affine spaces. Such constructions permit formally expressing the geometric picture of a velocity vector field as arrows tangential to a curve.