Vector Spaces

Abstract

Let F be a field. A nonempty set V together with a function (υ,υ’) ↦ υ + υ’ from V × V to V (called vector addition) and a function (a, υ) ↦ aυ from F × V to V (called scalar multiplication) is a vector space over the field F if and only if the following conditions are satisfied: (1) (associativity of vector addition) (υ1 + υ2) + υ3 = υ1 + (υ2 + υ3) for all υ1,υ2,υ3 ∊ V. (2) (commutativity of vector addition) υ1 + υ2 = υ2 + υ1 for all υ1, υ2 ∊ V. (3) (existence of an element neutral with respect to vector addition) There exists an element 0V of V satisfying υ + 0V = υ for all υ ∊ V. (4) (existence of inverses with respect to vector addition) For each element υ ∊ V there exists an element of V which we will denote by - υ and which satisfies υ + (-υ) = 0V. (5) (distributivity of scalar multiplication over vector addition) a(υ1 + υ2) = aυ1 + aυ2 for all a ∊ F and all υ1,υ2 ∊ V. (6) (distributivity of scalar multiplication over scalar addition) (a1 + a2)υ = a1υ + a2υ for all a1,a2 E F and all υ ∊V. (7) (associativity of scalar multiplication) a1(a2υ) = (a1(a2)υ for all a1,a2 ∊ F and all υ ∊ V. (8) (existence of an element neutral with respect to scalar multiplication) The element 1F ∊ F satisfies 1Fυ = υ for all υ ∊ V.