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

In summary, defining a vector space over Q is possible and all the axioms of a vector space should hold. This may be confusing for someone new to proofs, but it is similar to defining a vector space over R. Help is appreciated if anything is incorrect or unclear.
  • #1
Linday12
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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!
 
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  • #2
Since scalars are taken from a field, and Q is certainly a field, your definition makes sense.
 

1. What is a vector space over Q?

A vector space over Q is a mathematical structure that consists of a set of elements (vectors) that can be added together and multiplied by rational numbers. The set of rational numbers, denoted by Q, is the scalar field of the vector space.

2. How is a vector space over Q defined?

A vector space over Q is defined by a set of axioms, or properties, that must be satisfied. These axioms include closure under vector addition and scalar multiplication, associativity and commutativity of addition, and the existence of an additive identity and inverse elements.

3. Can a vector space only be defined over Q?

No, a vector space can be defined over any field. Q is just one example of a field, but vector spaces can also be defined over other fields such as R (real numbers) and C (complex numbers).

4. What are some examples of vector spaces over Q?

Some examples of vector spaces over Q include the set of all polynomials with rational coefficients, the set of all rational matrices, and the set of all infinite sequences of rational numbers.

5. Why is it important to define a vector space over Q?

Defining a vector space over Q allows us to study and manipulate mathematical objects using the rules and properties of vector spaces. This has many applications in various fields such as physics, engineering, and computer science.

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