IEVref: 102-03-01 ID: Language: en Status: Standard Term: vector space Synonym1: linear space [Preferred] Synonym2: Synonym3: Symbol: 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: $U+V=V+U$, $\left(U+V\right)+W=U+\left(V+W\right)$, 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 $\left(-U\right)$ such that $U+\left(-U\right)=0$, $\left(\alpha +\beta \right)\text{\hspace{0.17em}}U=\alpha \text{\hspace{0.17em}}U+\beta \text{\hspace{0.17em}}U$, where β is also a scalar, $\alpha \text{\hspace{0.17em}}\left(U+V\right)=\alpha \text{\hspace{0.17em}}U+\alpha \text{\hspace{0.17em}}V$, $\alpha \text{\hspace{0.17em}}\left(\beta \text{\hspace{0.17em}}U\right)=\left(\alpha \text{\hspace{0.17em}}\beta \right)\text{\hspace{0.17em}}U$, $1\text{\hspace{0.17em}}U=U$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. Publication date: 2017-07 Source: Replaces: 102-03-01:2007-08 Internal notes: CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: