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# vector space

There's night and day, brother, both sweet things; sun, moon, and stars, brother, all sweet things; there's likewise a wind on the heath. Life is very sweet, brother; who would wish to die?
George Borrow

## English

### Noun

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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 $\mathbb{F}$ is a one-dimensional vector space over itself.
• If $\mathbb{V}$ is a vector space over $\mathbb{F}$ and S is any set, then $\mathbb{V}^S=\{f|f:S\rightarrow \mathbb{V} \}$ is a vector space over $\mathbb{F}$, and $\mbox{dim} ( \mathbb{V}^S ) = \mbox{card}(S) \, \mbox{dim} (\mathbb{V})$.
• If $\mathbb{V}$ is a vector space over $\mathbb{F}$ then any closed subset of $\mathbb{V}$ is also a vector space over $\mathbb{F}$.
• The above three rules suffice to construct all vector spaces.

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