I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was $\Bbb{R} \times \Bbb{R}^3$, which gives us $(t,\mathbf{x})$. This space usually involves transformations of the form $g: (t,\mathbf{x}) \rightarrow (t + s,G\mathbf{x} + \mathbf{v}t + \mathbf{s})$. The particular space in question (Galilean space) can physically also be described by an affine space $A^4$ and my question was if there was an analog to $g$ on the affine space itself. But to answer this, seemingly I need to first establish if affine spaces can have coordinate transformations like the one I just did.
My Question
Can an affine space have coordinate transformations? If so, please provide an example.