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Vector space

Vector space



vector space
  1. 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.
    Each vector space has a basis and dimension.

vector space over the field F
  1. 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.