Are these rules "independent"? In other words, can I deduce some of these rules from others? If I want to verify if some collection of objects is a vector space, do I have to verify all these eight rules listed above, or only a subset of them would be sufficient?
How do we know that these rules are complete? It seems like we've gained them through observation, but how do they form a definition for a vector space? Is there a formal proof somewhere or does it just seem to work?
Are there any examples of collections that fulfill most (but not all) of these properties?