How to find a basis for the vector space of real numbers over the field Q?

[Arian.D]

So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers that are not algebraic. On the other hand we know from the axiom of choice that any vector space has a basis, so is there a way to find a basis for this interesting one?

I hope my question isn't wrong.

 

[Erland]

Probably, it is not possible to find such a basis without using at least some weaker form of the Axiom of Choice.

 

[Robert1986]

I might be way off here, but this is just a first thought. Have you heard of "the" (quotes because there is more than one) Vitali Set? The one I am thinking about is built like this: consider the interval [0,1]. Now, make an equivilence relation x~y iff x-y is rational. Now, pick one element from each equvilence class (you have to use the axiom of choice here.) This seems like it *might* form a basis for R over Q. But, like I said, this is one of the first things that popped in my mind so it might be wayyy off.

 

[Robert1986]

I think it might work. For example, if $[r]_{\alpha}$ is the collection of equivilence classes, and if $z \in [0,1]$ and $z \in [r]$ for some $r$ then $r-z = q \in \mathbb{Q}$. So that $z = 1r + q1$. So if we require that 1 be one of the numbers from the equivilency classes, then this set certainly spans $[0,1]$. And by taking $q$ to be an integer + q it seems like this set will span the whole real line. Now, are they linearly independent?

 

[blue_raver22]

I can confirm that your understanding of the vector space of real numbers over the field Q is correct. The dimension of this vector space is indeed infinite, as there are infinitely many real numbers that are not algebraic.

To find a basis for this vector space, we can use the axiom of choice to construct a basis using elements from the set of real numbers. This means that we can choose a set of linearly independent vectors that span the entire vector space. However, the exact construction of this basis may not be straightforward, as it depends on the specific elements chosen.

One possible approach to finding a basis for this vector space could be to use the concept of linear independence. We can start by choosing a set of real numbers that are linearly independent over the field Q. This means that no combination of these numbers using scalar multiplication and vector addition can result in the zero vector.

Once we have a set of linearly independent vectors, we can use the process of extending this set to a basis by adding in other vectors that are not in the span of the original set. This process can continue until we have a basis that spans the entire vector space.

In summary, while the exact construction of a basis for the vector space of real numbers over the field Q may not be straightforward, it is possible to use the axiom of choice and the concept of linear independence to find a basis for this interesting vector space.