A vector space is a set endowed with a rules for addition, and scalar multiplication. They must obey the following. For elements in the set f, g, and/or h.
1.(f+g)+h=f+(g+h) and c(kf)=(ck)f
This rule means that all elements in a vector space must obey the rules of association using both addition and multiplication.
2.f+g=g+f
This rule means that all elements in a vector space must obey the rules of commutation.
3.There exists one neutral element n in V such that f+n=f, which also denotes that n = 0.
4.There exists an element g in V such that f+g = 0. This denotes that for every element f there exists an element g which is equal to -f.
5.k(f+g)=kf+kg and (c+k)f=cf+kf
This rule means that all elements in a vector space must obey the rules of distribution.
6.1f=f
This ensures that a vector space obeys identity operations.
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