|Definition:|| for a given set of scalars (1), set of elements for which the sum of any two elements U and V and the product of any element and a scalar α are elements of the set, with the following properties: |
- , 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: ,
- there exists an opposite such that ,
- , where β is also a scalar,
NOTE 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 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.