A **vector space** over a [[field]] $F$ is - a set $V$ (which contains the [[Vector|vectors]]) - two closed [[binary operations]] which are closed over $V$ - [[vector addition]] $+: V \times V \rightarrow V$ - *Associativity* - $\mathbf{u}+(\mathbf{v}+\mathbf{w})=(\mathbf{u}+\mathbf{v})+\mathbf{w}$ - *Commutativity* - $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ - *Identity* - $\exists\mathbf0\in V\ :\ \mathbf{v+0=v}\qquad\forall\mathbf{v}\in V$ - *Inverse* - $\forall\mathbf{v}\in V\exists -\mathbf{v}\in V : v+(\mathbf{-v})=0$ - [[scalar multiplication]] $F \times V \rightarrow V$ - *Compatiblity* with the multiplication from $F$ - $a(b \mathbf{v})=(a b) \mathbf{v}$ - *Identity* - $1 \mathbf{v}=\mathbf{v}$, where 1 denotes the multiplicative identity in $F$ - *Distributivity* - Scalar to vector addition - $a(\mathbf{u}+\mathbf{v})=a \mathbf{u}+a \mathbf{v}$ - Vector to field addition - $(a+b) \mathbf{v}=a \mathbf{v}+b \mathbf{v}$