On Sun, 7 Nov 2004 15:52:58 +0000, Valentin Samko <boost@digiways.com> wrote:
If your teacher treats E^2 and R^2 differently, he is a one weird teacher. "In High School" is unrelated to a certain teacher, hence it doesn't really help when you call people weird. Also "is the same space" != "is treated as the same".
Also, if we do not treat E^n as a vector space, then the difference in this space is somewhat undefined. No. You are assuming difference in E^n is a binary operation, whereas it is a mapping into R^n.
In C++ sence - yes, it's a different point. In mathematical sense, point-point depends on how this operation is defined in your particular space, and in Euclidean space, the result of this operations is the same as difference of corresponding vectors. MW> Yes. But the result is a *vector*. IOW, the difference between 2 MW> points is a mapping from two points in Euclidean space to a vector in MW> a vector space: MW> difference :: E^n x E^n -> R^n I just do not get this. Why would you have a difference between two points in E^n defined as a point in R^n? MW> Please read what I wrote: ".. to a vector in a vector space.". R^n is MW> a *vector* space, hence the difference between two points in E^n is a MW> *vector* in R^n.
I probably used the wrong word there. By *point* I meant "an element", and element in R^n is a vector. You can not have "points" in R^n, which are not vectors. So we seem to agree.
Cheers, Michael