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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!

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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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  • $\begingroup$ If the associativity of addition fails for a space, then it is not a vector space. $\endgroup$ – Henricus V. Mar 28 '16 at 23:31
  • $\begingroup$ So, can you give me an example such that the associativity fails? Also, I am asking why associativity is so important so that it is one of the eight axioms? $\endgroup$ – Ivy Mar 29 '16 at 0:48
  • $\begingroup$ Have you tried doing linear algebra without ever using associativity? Try it and see if you think that property is important. It's hard to tell where you want to go with this. $\endgroup$ – David K Mar 29 '16 at 2:15
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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.

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