# Definition A [[set]] $V$ is a *vector space* (over a [[field|field]] $\mathbb{F}$) if it has a binary operation $+$ and (scalar) multiplication $\cdot$ by elements of the field, under which the set is closed, and the operations satisfy the following axioms: | | Property | Notation | |:---:|:--------------------------------------- |:--------------------------------------------------------------------:| | 1 | Commutativity | $\pmb{x} + \pmb{y} = \pmb{y} + \pmb{x}$ | | 2 | Assosciativity of vector addition | $(\pmb{x} + \pmb{y}) + \pmb{z} = \pmb{x} + (\pmb{y} + \pmb{z})$ | | 3 | Existence of the identity $\pmb{0}$ | $\pmb{x} + \pmb{0} = \pmb{x}$ | | 4 | Existence of the inverse $-\pmb{x}$ | $\pmb{x} + (-\pmb{x}) = \pmb{0}$ | | 5 | Distributivity over vector addition | $a \cdot(\pmb{x} + \pmb{y}) = (a \cdot \pmb{x}) + (b \cdot \pmb{y})$ | | 6 | Identity of scalar multiplication | $1 \cdot \pmb{x} = \pmb{x}$ | | 7 | Distributivity over field addition | $(a + b) \cdot \pmb{x} = (a \cdot \pmb{x}) + (b \cdot \pmb{x})$ | | 8 | Assosciativity of scalar multiplication | $(ab) \cdot \pmb{x} = a \cdot (b \cdot \pmb{x})$ | where $\pmb{x}, \pmb{y}, \pmb{z} \in V, \, a, b \in \mathbb{F}$. This is equivalent to saying $V$ is an [[Abelian group]] (with the binary operation $+$), which gives us axioms $1$ through $4$, and that scalar multiplication obeys axioms $5$ through $8$. Physicists usually only check axioms 3 and 4, the rest are quite natural.