| Definitions | | vector space |
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noun
- (maths) A type of set of vectors that satisfies a specific group of constraints.
- A vector space is a set of vectors which can be linear combination, linearly combined.
vector space over the field F
- (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
- :V→V, 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.
Translations: - Italian: spazio vettoriale
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