The **zero vector space** [[vector space]] whose only [[Element of a Set|element]] is [[zero vector]] $\set{\boldsymbol{0}}$ It is certainly the most simple vector space. Given $a,b\in F$, the zero vector space has - [[vector addition]] - *Associativity* - $\mathbf{0}+(\mathbf{0}+\mathbf{0})=(\mathbf{0}+\mathbf{0})+\mathbf{0}$ - *Commutativity* - $\mathbf{0}+\mathbf{0}=\mathbf{0}+\mathbf{0}$ - *Identity* - $\exists\mathbf0\in \set{\mathbf0}\ :\ \mathbf{0+0=0}\qquad\forall\mathbf{0}\in \set{\mathbf0}$ - *Inverse* - $\forall\mathbf{0}\in \set{\mathbf0}\exists (\mathbf0)\in \set{\mathbf0} : \mathbf{0}+(\mathbf{0})=0$ - [[scalar multiplication]] $F \times \set{\mathbf0} \rightarrow \set{\mathbf0}$ - *Compatiblity* with the multiplication from $F$ - $a(b \mathbf{0})=(a b) \mathbf{0}$ - *Identity* - $1 \cdot\mathbf{0}=\mathbf{0}$, where 1 denotes the multiplicative identity in $F$, but literally any number works - *Distributivity* - Scalar to vector addition - $a(\mathbf{0}+\mathbf{0})=a \mathbf{0}+a \mathbf{0}$ - Vector to field addition - $(a+b) \mathbf{0}=a \mathbf{0}+b \mathbf{0}$