# Vector Spaces

 .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } (VS8) Distributive property of scalar multiplication over scalar addition (VS7) Distributive property of scalar multiplication over vector addition (VS6) Associativity of scalar multiplication (VS5) Unit Property (VS4) Existence of additive inverses in V (VS3) Existence of a zero vector (VS2) Associativity of addition (VS1) Commutativity of addition (C2) Closure under scalar multiplication For each vector v∈V and each scalar a∈F, the scalar multiple av belongs to V (C1) Closure under addition For each pair of vectors u,v∈V, the sum u+v also belongs to V Definition of a Vector Space Let V be a nonempty set and let F be a field.  Suppose that an addition operation and a scalar multiplication operation are defined on V, with scalars belonging to the field F.  We call V a vector space over F provided that satisfies: Closure under additionClosure under scalar multiplicationCommutativity of additionAssociativity of additionExistence of a zero vectorExistence of additive inverses in VUnit PropertyAssociativity of scalar multiplicationDistributive property of scalar multiplication over vector additionDistributive property of scalar multiplication over scalar addition (F5) Distributivity of multiplication over addition a·(b+c)=a·b+a·c (F4) Inverses for addition and multiplication For each element a in F and each nonzero element b in F, there exist elements c and d in F such that a+c=0 and b·d=1 (F3) Additive and multiplicative identity elements There exist distinct elements 0 and 1 in F such that 0+a=a and 1·a=a (F2) Associativity of addition and multiplication (a+b)+c=a+(b+c) and (a·b)·c=a·(b·c) (F1) Commutativity of addition and multiplication a+b=b+a and a·b=b·a Define a Field Commutativity of addition and multiplicationAssociativity of addition and multiplicationAdditive and multiplicative identity elementsInverses for addition and multiplicationDistributivity of multiplication over addition .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } Authorlazvertiigo ID333670 Card SetVector Spaces DescriptionVector Spaces Updated2017-08-23T16:07:26Z Show Answers