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Given a field \((k,+,\times,0_k,1_k)\), a 'vector space over \(k\)', or a '\(k\)-vector space' \((V,+,\cdot)\) is a set \(V\), with a binary operation \(+:V\times V\to V\) ('vector addition') such that \((V,+)\) is an abelian group, and an operation \(\cdot:k\times V\to V\) ('scalar multiplication') which is left-unital (i.e. \(\forall v\in V:1_k\cdot v=v\)), associates with multiplication in the field (i.e. \(\forall a,b\in k:\forall v\in V:a\cdot(b\cdot v)=(a\times b)\cdot v\)), and distributes over both scalar and vector addition (\(\forall a,b\in k:\forall u,v\in V:(a+b)\cdot(u+v)=a\cdot u+a\cdot v+b\cdot u+b\cdot v\)).