Vector spaces seem to have a very similar definition to fields. Are vector spaces fields?
Thanks.
Vector spaces seem to have a very similar definition to fields. Are vector spaces fields?
Thanks.
No. Vector spaces are not fields. Look at the definition of the vector space. First you consider a group $(V,+)$ which is commutative. Then you consider a field $\Bbb{K}$ whose operations are compatible to those of $V$, i.e.
$ (\alpha+\beta)x=\alpha x+\beta x,\ \forall x \in V,\alpha,\beta \in \Bbb{K}$
and so on. Notice that the scalar multiplication does not take place between two elements of $V$, but between ONE element of $\Bbb{K}$ and ONE element of $V$.
As a quick example, you could look at $\Bbb{R}^3=\{ (x,y,z) : x,y,z \in \Bbb{R}\}$ with addition $(a,b,c)+(d,e,f)=(a+d,b+e,c+f)$ and scalar multiplication $\alpha(a,b,c)=(\alpha a,\alpha b,\alpha c)$. This is a vector space over $\Bbb{R}$ (you can check all the axioms). But $\Bbb{R}^3$ is not a field: the product $(a,b,c)\cdot(d,e,f)$ is not defined. The product is defined between one scalar and one vector.
I hope this makes things a little bit clear. :)