Vector space
The set
\(V\) defined over the field \(F\)
that contains the zero vector, so
\begin{equation*} \vec{0} \in V, \end{equation*}in conjunction with the vector addition operation
\begin{equation*} \forall \vec{u}, \vec{v} \in V \> : \> \vec{u} + \vec{v} \in V \end{equation*}
that satisfies all properties listed here
and scalar multiplication operation
\begin{equation*} \forall a \in F, \> \forall \vec{v} \in V \> : \> a \vec{v} \in V \end{equation*}
that satisfies all properties listed here.
which is the one and only
link between vector addition and scalar multiplication
in the vector space.