A vector space is an algabraic structure over a [[Field]] $F$ is a [[Empty Set|nonempty]] [[Set|set]] $V$ combined with a [[Binary Operation|binary operation]] $+$ and a [[Arity|binary]] [[Function|function]] (denoted as $ab$ with no symbol for the operator between $a$ and $b$) such that the following [[Axiom|axioms]] hold: For any [[Scalar|scalar]] in the [[Field|field]] $a,b \in F$ and any [[Vector|vector]] $\vec v,\vec u, \vec w \in V$: - The [[Binary Operation|binary operation]] $+$ and the [[Set|set]] of vectors $V$ must form an [[Abelian Group]] $[V,+]$ where we denote its [[Identity Property|identity]] as the [[Zero Vector]] $\vec 0$. - Multiplication in the [[Field|field]] $F$ must be [[Associative Property|Associative]] with the binary function. $\huge (ab)\vec v = a(b\vec v) $ - The [[Identity Function|identity]] $1$ of multiplication in $F$ must be the identity of the binary function $ \huge 1\vec v = \vec v $ - [[Scalar]] multiplication in $F$ must be [[Distributive Property|distributive]] over the [[Binary Operation|binary operation]] $+$ (and vice-versa) $\huge \begin{align} a(\vec u + \vec v) &= a\vec u + a\vec v \\ (a+b)\vec v &= a \vec v + b\vec v \end{align}$ A common construction of a vector space is one over field of [[Real Numbers]] such that [[Scalar|Scalars]] are just a real number over finite-dimensioned [[Vector|vectors]] that are list of [[Scalar|scalars]].