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vector space

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English

Noun

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vector space

  1. (mathematics) A type of set of vectors that satisfies a specific group of constraints.
    A vector space is a set of vectors which can be linearly combined.

vector space over the field F

  1. (linear algebra) A set V, whose elements are called "vectors", together with a binary operation + forming a module (V,+), and a set F* of bilinear unary functions f*:VV, each of which corresponds to a "scalar" element f of a field F, such that the composition of elements of F* corresponds isomorphically to multiplication of elements of F, and such that for any vector v, 1*(v) = v.
    • Any field <math>\mathbb{F}</math> is a one-dimensional vector space over itself.
    • If <math>\mathbb{V}</math> is a vector space over <math>\mathbb{F}</math> and S is any set, then <math>\mathbb{V}^S=\{f|f:S\rightarrow \mathbb{V} \}</math> is a vector space over <math>\mathbb{F}</math>, and <math> \mbox{dim} ( \mathbb{V}^S ) = \mbox{card}(S) \, \mbox{dim} (\mathbb{V})</math>.
    • If <math>\mathbb{V}</math> is a vector space over <math>\mathbb{F}</math> then any closed subset of <math>\mathbb{V}</math> is also a vector space over <math>\mathbb{F}</math>.
    • The above three rules suffice to construct all vector spaces.

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