I recently bought a new linear algebra book. I've been through the subject before, but the book I had back then glossed over the abstract details of vector spaces, amongst other things. However, I've found something questionable in the first few pages, and I was wondering if someone could give me thier opinion on. It states:

I've no problem with any of this, but I have never seen any mention of this distinction before. This is compounded by the fact that Wikipedia seems to use the two spaces interchangeably in a context of vectors, and MathWorld seems to say that [tex]\mathbb{E}^n[/tex] is just an older notation for [tex]\mathbb{R}^n[/tex]. Is this just a case of people carelessly confusing the two, or is this book promoting nonsense?

This is the distinction between and "Affine" space and a "Vector" space. For example, in E^{2}, we can talk about lines through points and the distance between points but we do not add points or multiply points by numbers.

Of course, as soon as we set up a coordinate system in a plane, we can, as in basic calculus, talk about the vector from 0 to a point and so associate a vector with a point. Then it becomes R^{2}.