Any specific reasons for defining a vector space to have the properties like closed under addition, closed under multiplication, associative, distributive...Why only these properties? (What makes these properties necessary and sufficient?)
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$\begingroup$ These properties make linear combination possible. $\endgroup$– Henricus V.Nov 14, 2015 at 4:46
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The ten axioms that a set of vectors must satisfy in order to be considered a vector space are necessary in order to have a mathematical model that can actually be applied to the real world.
For instance, if you have the vector space consisting of all polynomials, it wouldn't make any sense to add two polynomials (two vectors in that space) and have their sum be a member of a different space, say the set of all two by two matrices.
As to why those axioms are sufficient, they were just determined to be that way by whoever was coming up with the definition of a vector space. Definitions that didn't contain all of these axioms, on the other hand, weren't sufficient and didn't provide adequate models. They were found to be inconsistent somehow.
