Vector Space

A set V equipped with two operations (vector addition and scalar multiplication by elements of a field) satisfying axioms: associativity, commutativity of addition, identity and inverse for addition, distributivity of scalar multiplication over field addition and vector addition, compatibility of scalar multiplication, and identity scalar acting as identity; vectors can be combined linearly.

An algebraic structure V over a field F consisting of an abelian group (V, +) and a scalar multiplication F × V → V satisfying distributivity, associativity with field multiplication, and identity scalar action; vectors can be added and scaled by field elements.

A set V equipped with an abelian group operation + and a scalar multiplication by elements of a field F, satisfying distributivity, associativity of scalar multiplication, and 1·v = v; equivalently, a module over a field.

A collection of objects called vectors forming an abelian group under addition together with scalar multiplication by elements of a field such that the usual axioms (distributivity, associativity, identity) hold; equivalently, a module over a field.