From: Valentin Samko (boost_at_[hidden])
Date: 2004-11-07 10:52:58
>> MW> The Euclidean space is E^3, the vector space is R^3.
>> MW> Please see: http://www.ma.umist.ac.uk/kd/curves/node3.html
>> MW> Note also: "Not all books make this distinction so you need to be
>> MW> prepared to encounter the unstated identification E^3 = R^n"
>> MW> Mathworld is using this _unstated_ _identification_; it is imprecise at least.
>> It's not just Mathworld, it's thousands of books/articles. In any
>> case, don't we assume, the gui library will use R^2 space?
MW> Most of the books/articles I have seen so far regard points as
MW> something different than vectors. It is even the way it's taught in
MW> High School ;-)
If your teacher treats E^2 and R^2 differently, he is a one weird
teacher. Nowadays everyone considers them as the same space.
If you google for Euclidean+vector+space, you will find plenty of
lecture notes, etc, which assume that E^n is the same space as R^n,
and element in E^n is a vector.
Considering the link you posted - either that guy is trying to be weird,
or what he states there is only related to how he teaches the curve
theory. Besides, he is being inconsistent in his definitions.
Also, if we do not treat E^n as a vector space, then
the difference in this space is somewhat undefined.
MW> Usually you will read something like "a point can be
MW> specified/represented by a vector" (identification E^3=R^3).
MW> Everything else uses abovementioned identification, which, when
MW> unstatedly used, is at least imprecise.
>> >> 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.
See http://mathworld.wolfram.com/VectorSpace.html .
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