I have to find an example (if there exists one) of a field $F$ and a structure $(V,+,s)$ (where $s$ stands for scalar multiplication) satisfying all but one of the axioms of a vector space, namely the distributivity of vector addition: For all $c ∈ F$ and $v,w ∈ V$ , $c(v + w) = cv + cw$

I believe such example exists, but I'm not sure where to start. An exercise I did before asked to show how the axiom follows from the other axioms if $F=\mathbb{Q}$. So I think perhaps using a different field like $\mathbb{R}$ (or anything similar) could be of use.

Could anyone give me some advice or
hint to solve it?


  • $\begingroup$ "namely the distributivity of vector addition." - To be clear, is this the axiom you mean? ($a \in F; \vec v, \vec w \in V$; scalar multiplication denoted by $\cdot$): $$a \cdot (\vec v + \vec w) = a \cdot \vec v + a \cdot \vec w$$ $\endgroup$ Aug 6, 2020 at 2:32
  • $\begingroup$ I think so. I see none other. $\endgroup$
    – user732848
    Aug 6, 2020 at 2:33
  • $\begingroup$ Yes, that's the axiom $\endgroup$ Aug 6, 2020 at 2:38
  • 1
    $\begingroup$ Maybe math.stackexchange.com/questions/1412899/… will do. $\endgroup$ Aug 6, 2020 at 4:15
  • 1
    $\begingroup$ See math.stackexchange.com/a/479005/589 $\endgroup$
    – lhf
    Aug 6, 2020 at 10:23


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