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A set of vectors V, associated with a field of scalars F, with the following properties:
1) There is an operation of vector addition, which is associative and commutative and has an identity element 0 which is in V, and if a and b are in V, then a+b is in V; also the additive inverse of any vector in V is in V;
2) There is an associative operation which multiplies a vector by a scalar; multiplication is distributive over both vector addition and scalar addition; if v is in V and f is in F then f*v is in V; multiplication by the unit element of F leaves elements of V unchanged.

An Abelian group (V,+) and a map m : F ×V → V where F is a given field such that, writing λv instead of m( λ,v) , we have
1v=v ,
( λμ)v = λ( μv) ,
( λ+ μ)v = λv+ μv ,
λ(v+w) = λv+ λw
for all λ, μ ∈ F and all v,w ∈ V . We say that V is a vector space over F.
See also dual, dimension, basis, normed vector space.

An Abelian group (V, +), with a field F and a binary operation combining a member of F with a member of V to give another member of V.
The members of V are called vectors and those of F are called scalars. The binary operation connecting the two is called scalar multiplication.