Exploring Spherical Geometry: A Question on Vector Space and Basis Formation

In summary: You could also define "add" and "multiply" in terms of "rotate" and "scale". For example, "add" could be defined as rotating the first vector by the second vector about the vector's origin. "Multiply" could be defined as rotating the first vector about the second vector's origin and scaling it by the second vector's magnitude.
  • #1
crownedbishop
22
1
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?
 
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  • #2
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.
 
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  • #3
crownedbishop said:
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.
 
  • #4
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.
 
  • #5
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).
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects called vectors, which can be added together and multiplied by scalars (numbers). This structure is used to describe and analyze the properties of vectors and their operations.

2. What are the basic properties of a vector space?

The basic properties of a vector space include closure under vector addition and scalar multiplication, associativity and commutativity of addition, distributivity of scalar multiplication over vector addition, and the existence of an additive identity element and scalar identity element.

3. How do you determine if a set is a vector space?

To determine if a set is a vector space, you need to check if it satisfies all the basic properties of a vector space. This includes checking if it is closed under vector addition and scalar multiplication, if the operations are associative and commutative, and if there exist identity elements for both operations.

4. What is the difference between a vector space and a subspace?

A vector space is a set of objects that satisfies all the basic properties of a vector space, while a subspace is a subset of a vector space that also satisfies these properties. In other words, a subspace is a smaller vector space contained within a larger vector space.

5. How are vector spaces used in real-world applications?

Vector spaces have many applications in various fields, including physics, engineering, computer science, and economics. They are used to model and analyze physical phenomena, such as forces and motion, as well as to represent data and perform calculations in computer programs and algorithms.

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