# Vector spaces

1. Oct 17, 2009

### kman12

Hello, I'm studying linear algebra and wanted to know what is the difference between a "vector space" and a "vector space over field F".
I know that a vector space over field F satisfies the 8 axioms, but does a vector space satisfy this also?

2. Oct 17, 2009

### waht

A field could be a set of real, or complex numbers, or a set of rational numbers whose addition and multiplication is well defined.

3. Oct 17, 2009

### kman12

This doesn't answer my question, I want to know the difference between a vector space and a vector space over field F.

4. Oct 17, 2009

### Fredrik

Staff Emeritus
There's no difference. Every vector space is a vector space over a field. The field is usually the real numbers or the complex numbers, but it could be any field. A vector space over $\mathbb R$ is often called a "real vector space", and a vector space over $\mathbb C$ is often called a "complex vector space".

5. Oct 17, 2009

### kman12

Right this would make sense. So that means the 8 axioms for a "Vector Space over a field F" also hold for a "Vector space".

Because i know that the basic defin of a vector space is that:
1) It contains a non empty set V whose elements are vectors
2) A field F whose elements are scalars
3) A binary operation + on V Under which V is closed
4) A multiplication . of a vector by a scalar.
So on top of this the 8 axioms (That hold for a vector space over a field F) also hold for a vector space (i cant be asked to write all axioms)?

6. Oct 17, 2009

### Fredrik

Staff Emeritus
Yes, you got it right. I prefer to state the definition a bit differently though. In my definition, V is a set, $\mathbb F$ is a field, and $A:V\times V\rightarrow V$ and $S:\mathbb F\times V\rightarrow V$ are functions (called "addition" and "scalar multiplication" respectively). We use the notation $A(x,y)=x+y$ and $S(k,x)=kx$.

Definition: A 4-tuple $(V,\mathbb F,A,S)$ is said to be a vector space over the field $\mathbb F$ if

(i) $(x+y)+z=x+(y+z)$ for all $x,y,z\in V$

...and so on. (You seem to know the rest).

Note that V is just a set. It's convenient to call V a vector space, but you should be aware that this is actually a bit sloppy. It's certainly OK to do it when it's clear from the context what field $\mathbb F$ and what addition and scalar multiplication functions we have in mind. For example, it's common to refer to "the vector space $\mathbb R^2$ " because everyone is familiar with the standard vector space structure on that set.

7. Oct 17, 2009

### wofsy

I find it useful to think of vector spaces as special cases of modules over rings.
A module is just an abelian group together with a distributive multiplication by elements of a ring. If the ring is a field then the module is a vector space.

The distinguishing feature of a field is that is has multiplicative inverses.

Much of the theory of vector spaces actually comes from considering modules where the ring is a principal ideal domain. This is because the ring of polynomials over a field is a principal ideal domain.

Last edited: Oct 17, 2009
8. Oct 18, 2009

### kman12

right thanks fredrik