E^n vs R^n

• Sep 16th 2009, 08:28 AM
tjkubo
E^n vs R^n
What is the difference between E^n and R^n? My professor uses these often and I can't seem to distinguish between them, except somehow I think E^n is more general.
• Sep 16th 2009, 10:44 AM
HallsofIvy
"$\displaystyle E^n$" is the set of all points in n-dimensional Euclidean space. For example, $\displaystyle E^2$ is the two dimensional plane. "$\displaystyle R^n$", on the other hand, is that set with a specific point as "origin" and a given coordinate system. That way, $\displaystyle R^n$ can be written as a set of ordered "n-tuples", $\displaystyle \{x_1, x_2, \cdot\cdot\cdot, x_n\}$, has a dot-product defined on it, and allows you to define angles.
• Sep 16th 2009, 10:45 AM
Matt Westwood
Quote:

Originally Posted by tjkubo
What is the difference between E^n and R^n? My professor uses these often and I can't seem to distinguish between them, except somehow I think E^n is more general.

Depends on the context, but I'm prepared to make a guess ...

$\displaystyle \mathbb{R}^n$ could well be the general n-dimensional real vector space: when n=1 you have the number line, n=2 you have the plane, n=3 you have, er, space ... it can be shown (but not by me, I'm a duffer at such things) that this is isomorphic to these spaces are isomorphic to what they are given to represent, which is why (given an appropriate frame of reference) any point in space can be mapped 1-1 to $\displaystyle \mathbb{R}^3$ so that every point can be identified by an ordered triple of 3 real numbers

As for $\displaystyle \mathbb{E}^n$ ... now I'm assuming that here $\displaystyle \mathbb{E}^n$ is your general n-dimensional Euclidean space:
Definition:Euclidean Space - ProofWiki
... but the context is unclear, so as I say, I'm guessing.

Whoops, HallsOfIvy beat me to it.
• Sep 16th 2009, 12:35 PM
tjkubo
Wait, I'm confused. You guys' definitions seem to contradict each other. HallsOfIvy is saying E^n is like R^n, except there is no defined origin or coordinates. But Matt's link says E^n is the set R^n with a metric defined. Maybe I'm misinterpreting. Are there more than one uses for these terms depending on the context? Can you guys recommend a good math book that defines these terms clearly?
• Sep 16th 2009, 01:25 PM
Matt Westwood
I learned what I know from W. A. Sutherland's "Introduction to Metric and Topological Spaces", if that helps ...

To put it into context: $\displaystyle \mathbb{R}^n$ is (technically) just the Cartesian product of n copies of the real number line, with no added "structure".

However, $\displaystyle \mathbb{E}^n$ is that same $\displaystyle \mathbb{R}^n$ with the added concept of "distance between points", defined in the same way as real-world distances. It does the job of filling in the (otherwise intuitive) concept of spatial relationships between objects in that space. Thus $\displaystyle \mathbb{E}^n$ does a better mathematical job, if you like, of defining space, and is actually closer to a mathematical definition of what space "actually is".

As HallsOfIvy says, by defining the "dot product" on two vectors in this space-with-a-metric, you then get the concept of angle.

Ask your professor to define the difference between the two, he may do a better job of it than me.
• Sep 16th 2009, 01:45 PM
HallsofIvy
Not exactly "contradicting" but my definition of $\displaystyle R^n$ does require more that just having a metric- i.e. being able to measure lengths. I am requiring that we be able to measure angles as well.
• Sep 16th 2009, 01:52 PM
Matt Westwood
Okay, you've made me have to go away and think about whether by assigning a metric to $\displaystyle R^n$ you can deduce the concept of angle without requiring any further structure (i.e. "angle") to be imposed ... I had a feeling that the concept of angle could be deduced elementarily from the concept of the space with the Euclidean metric imposed. This needs thinking about ...