| Definition: | for a given set of scalars, 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: - 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,
- (α+β) U=α U+β U, where β is also a scalar,
- α (U+V)=α U+α V,
- α (β U)=(α β) U,
- 1 U=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.
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