For example, at: https://en.wikipedia.org/wiki/Vector_space
There are 8 axioms that a qualify a set to be a vector space. My professor also gave us 8.
However, a textbook I'm reading states 10 axioms. It has all 8 of the above mentioned axioms, as well as the following two:
If $u$ and $v$ are in $V$, then $u+v$ is also in $V$
If $u$ is in $V$, then $cu$ is in V (where $c$ is an arbritrary scalar)
Why is it that these two axioms aren't explicitly stated in some places? As far as I can't tell it's not that they're included within the 8 existing axioms. I know that these are the requirements for something to qualify as a subspace of a vector space...does that have anything to do with it?
I'd just like to know why these two axioms aren't explicitly stated as they seem very important and don't seem to be indirectly covered by any of the other axioms. Any help is appreciated!