Question on vector space

[crownedbishop]

So I was thinking and I was wondering if we could have a set of vectors that spanned just the unit sphere, and nothing else beyond that. So, if we replace euclid's 5th postulate to give us spherical geometry, a line is a circle on the surface of some sphere. If we have two perpendicular vectors (or two linearly independent) vectors in our spherical geometry, then it would seem that the whole circle of radius r is spanned. If we combine this and do it for all real numbers x such that 0<x<r and add the orgin, we will get a basis for the whole sphere which is a subset of the euclidean plane. Alternatively, I can imagine we could've done the same thing with complex multiplication. I was wondering:
1) Does this actually work?
2) If it does work, what kind of geometrical shapes can you form a basis for?

 

[Fredrik]

It doesn't work. Every non-empty subset of $\mathbb R^3$ spans a subspace of $\mathbb R^3$ that includes points that aren't on the unit sphere.

 

[Stephen Tashi]

So I was thinking and I was wondering if we could have a set of vectors that spanned just the unit sphere, and nothing else beyond that.

Are you are asking if you can represent a sphere by using the usual method of treating vectors as 3-tuples of numbers? No, that wouldn't work. The sum of two vectors on the sphere would be off the sphere, so the set of vectors on the unit sphere wouldn't satisfy the axiom a vector space that says the sum of two vectors in the space must also be in the space.

However, the mathematical definition of a vector space is more general than the usual way of dealing 3-tuples of numbers. It would interesting to see what you can come up with.

A general approach to represent a surface as a set of vectors would be to find a 1-to-1 mapping F from the surface to the 2-D plane. Then for points P and Q on the surface, you define the operation P+Q to be: Use F to map P and Q to vectors in the 2-D plane. Do the addition on the vectors in the 2-D plane the usual way. Then map the answer back to the surface by using the inverse function of F.

This might not be the kind of thing you're looking for. However, it is mathematically legal to define the addition of vectors in a vector space in a complicated way, as long as the addition satisifies the mathematical axioms.

 

[mathkid]

That's an interesting question. I'll think more about it tomorrow, but maybe you'll find it interesting what I thought so far. I was thinking of a way to associate R3 with the sphere, but some points had to be removed. Let a plane in R3 map to the riemann sphere of radius x such that 0<x<r. Consider parallel planes that map to the riemann sphere such that as you go in one direction, the planes map to a riemann sphere approaching radius r, and in the other direction, the planes map to a riemann sphere approaching radius 0. Our result is a unit ball with a line segment removed, that is from the North Pole to the orgin. If we associate each vector in R3 with a new "vector" in our modified ball, the vector space axioms should fit the bill.

 

[Svein]

Well, you "can" do it, but you need to come up with new definitions of "add" and "multiply". Not to worry, relativistic theory has already done that (adding two velocities close to c results in a new velocity closer to c, but not exceeding it).