Understanding Vector Spaces in Linear Algebra

[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 their opinion on. It states:

To clearly identify \mathbf{p} as a point, the notation \mathbf{p} \in \mathbb{E}^n is used.
...
To clearly identify \mathbf{v} as a vector, we write \mathbf{v} \in \mathbb{R}^n
...
Points and vectors are different geometric entities. This is reiterated by saying they live in different spaces, \mathbb{E}^n and \mathbb{R}^n... The primary reason for differentiating between points and vectors to achieve geometric constructions which are coordinate independent.

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?

 

[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.

 

[Parthalan]

Perfect! Thanks, HallsofIvy.