From: Michael Walter (michael.walter_at_[hidden])
Date: 2004-11-07 15:00:23
On Sun, 7 Nov 2004 15:52:58 +0000, Valentin Samko <boost_at_[hidden]> wrote:
> If your teacher treats E^2 and R^2 differently, he is a one weird
"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.