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vector space

A set with addition and scalar multiplication obeying ten algebraic axioms.

A vector space is a set V with an addition rule and a scalar multiplication rule that never leave V, satisfying ten axioms: closure under both operations, commutativity and associativity of addition, a zero vector, additive inverses, and the distributive/associative laws linking scalars and vectors. \mathbb R^2 satisfies all ten with ordinary coordinate arithmetic, but the definition says nothing about arrows or coordinates — any set with operations obeying the ten promises earns the name, including matrices, polynomials, and functions.

See also

subspace basis dimension
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