Defining a Vector Space over Q: Can It Be Done?

[Linday12]

Homework Statement

I'm trying to define a vector space over Q. Does this make any sense?

Homework Equations

The properties of a vector space

The Attempt at a Solution

Let V=Q^2 over Q. It seems to me that everything would be defined and I shouldn't be able to do anything to a vector in this space to make it become an irrational or anything else that could let it step outside the field, so all the axioms of a vector space should hold.

This is the first class I've taken that actually deals with proofs, and I'm not following along too well. I was just wondering if you could do this. It seems quite similar to something like, Let V=R^2 over R.

So if I have something wrong here or make completely no sense, help would be appreciated. Thank you!

 

[radou]

Since scalars are taken from a field, and Q is certainly a field, your definition makes sense.