• $U+V=V+U$,
• $(U+V)+W=U+(V+W)$, where W is also an element of the set,
• there exists a neutral element for addition, called zero vector and denoted by 0, such that: $U+0=U$,
• there exists an opposite $(-U)$ such that $U+(-U)=0$,
• $(\alpha +\beta )U=\alpha U+\beta U$, where β is also a scalar,
• $\alpha (U+V)=\alpha U+\alpha V$,
• $\alpha (\beta U)=(\alpha \beta )U$,
• $1U=U$

Note 1 to entry: In the usual three-dimensional space, the directed line segments with a specified origin form an example of a vector space over real numbers. Another example, corresponding to the extended concept of scalar (see IEV 102-02-18, Note 1) is the set of n-bit words formed of the digits 0 and 1 with addition modulo two, where the set of scalars is the set of two elements 0 and 1 subject to Boolean algebra.