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?

gfkang

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?

willowpet

Are there any examples of collections that fulfill most (but not all) of these properties?

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?