If you have learned anything about linear algebra, checking whether a space is a vector space is the most fundamental task we must do in the first class. However, it always makes me feel tedious to do so. Therefore, I am asking for some cool or bizarre examples or some further explanations about what actually a vector space is to help me understand deeper what I am actually doing with checking the eight axioms. For example, associativity under addition is always trivial. But I can not imagine some spaces without associativity, or why associativity is so important ? Thank you!
closed as off-topic by Henricus V., Adam Hughes, Daniel W. Farlow, David K, Alex Provost Mar 29 '16 at 2:38
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The axioms of vector space is related to definition of operation. So, if you define the addition as subtraction then it is not associative,$$a-(b-c)\neq (a-b)-c$$ there are many operation that are not associative. But it should be noted that if a vector space is given, all its subsets (note! subsets) have associativity and other properties, and you have to check only closeness to addition and scalar multiplication to be subspace.