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IEVref: | 102-03-01 | ID: | |

Language: | en | Status: backup | |

Term: | vector space | ||

Synonym1: | linear space [Preferred] | ||

Synonym2: | |||

Synonym3: | |||

Symbol: | |||

Definition: | for a given set of scalars (1), set of elements for which the sum of any two elements and U and the product of any element and a scalar Vα are elements of the set, with the following properties: - $U+V=V+U$,
- $(U+V)+W=U+(V+W)$, where
is also an element of the set,*W* - 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 $U+(-U)=0$,
- $(\alpha +\beta )\text{\hspace{0.17em}}U=\alpha \text{\hspace{0.17em}}U+\beta \text{\hspace{0.17em}}U$, where
*β*is also a scalar, - $\alpha \text{\hspace{0.17em}}(U+V)=\alpha \text{\hspace{0.17em}}U+\alpha \text{\hspace{0.17em}}V$,
- $\alpha \text{\hspace{0.17em}}(\beta \text{\hspace{0.17em}}U)=(\alpha \text{\hspace{0.17em}}\beta )\text{\hspace{0.17em}}U$,
- $1\text{\hspace{0.17em}}U=U$
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 | ||

Publication date: | 2007-08 | ||

Source: | |||

Replaces: | |||

Internal notes: | |||

CO remarks: | |||

TC/SC remarks: | |||

VT remarks: | |||

Domain1: | |||

Domain2: | |||

Domain3: | |||

Domain4: | |||

Domain5: |

- $U+V=V+U$,
- $(U+V)+W=U+(V+W)$, where
is also an element of the set,*W* - 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 $U+(-U)=0$,
- $(\alpha +\beta )\text{\hspace{0.17em}}U=\alpha \text{\hspace{0.17em}}U+\beta \text{\hspace{0.17em}}U$, where
*β*is also a scalar, - $\alpha \text{\hspace{0.17em}}(U+V)=\alpha \text{\hspace{0.17em}}U+\alpha \text{\hspace{0.17em}}V$,
- $\alpha \text{\hspace{0.17em}}(\beta \text{\hspace{0.17em}}U)=(\alpha \text{\hspace{0.17em}}\beta )\text{\hspace{0.17em}}U$,
- $1\text{\hspace{0.17em}}U=U$

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.

102-03-01en.gif |