# E^n vs R^n

1. Jan 29, 2007

### Parthalan

Hi,

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 $$\mathbb{E}^n$$ is just an older notation for $$\mathbb{R}^n$$. Is this just a case of people carelessly confusing the two, or is this book promoting nonsense?

Last edited: Jan 29, 2007
2. Jan 29, 2007

### HallsofIvy

This is the distinction between and "Affine" space and a "Vector" space. For example, in E2, 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 R2.

3. Jan 29, 2007

### Parthalan

Perfect! Thanks, HallsofIvy.