Do vector spaces have multiplicative inverses?
They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$
And it is even a group under addition it would seem, but without that inverse on multiplication, we can't have a group under multiplication, and thus not a ring right?
Best Answer
To say that $G$ is a group under multiplication means that it is possible to multiply elements of $G$ by elements of $G$ in such a way that the group axioms are satisfied.
In vector spaces you do not multiply vectors by vectors, you multiply vectors by scalars, so you don't even get started on groups.
It is true that in some cases you can multiply vectors by vectors, but this is not a part of vector space theory. For example (exercise) you might like to investigate whether or not the set of all real polynomials forms a group under ordinary polynomial multiplication. Or whether $\Bbb R^3$ forms a group under the vector cross product. Or whether the set of all (be careful!) $2\times2$ matrices forms a group under matrix multiplication. Once again remember though, this is not a question about vector spaces.