Vector Spaces

  1. (VS8) Distributive property of scalar multiplication over scalar addition
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  2. (VS7) Distributive property of scalar multiplication over vector addition
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  3. (VS6)
    Associativity of scalar multiplication
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  4. (VS5) Unit Property
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  5. (VS4) Existence of additive inverses in V
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  6. (VS3) Existence of a zero vector
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  7. (VS2) Associativity of addition
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  8. (VS1) Commutativity of addition
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  9. (C2) Closure under scalar multiplication
    For each vector v∈V and each scalar a∈F, the scalar multiple av belongs to V
  10. (C1) Closure under addition
    For each pair of vectors u,v∈V, the sum u+v also belongs to V
  11. 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 addition
    • Closure under scalar multiplication
    • Commutativity of addition
    • Associativity of addition
    • Existence of a zero vector
    • Existence of additive inverses in V
    • Unit Property
    • Associativity of scalar multiplication
    • Distributive property of scalar multiplication over vector addition
    • Distributive property of scalar multiplication over scalar addition
  12. (F5) Distributivity of multiplication over addition
    a·(b+c)=a·b+a·c
  13. (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
  14. (F3) Additive and multiplicative identity elements
    There exist distinct elements 0 and 1 in F such that

    0+a=a and 1·a=a
  15. (F2) Associativity of addition and multiplication
    (a+b)+c=a+(b+c) and (a·b)·c=a·(b·c)
  16. (F1) Commutativity of addition and multiplication
    a+b=b+a and a·b=b·a
  17. Define a Field
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    • Commutativity of addition and multiplication
    • Associativity of addition and multiplication
    • Additive and multiplicative identity elements
    • Inverses for addition and multiplication
    • Distributivity of multiplication over addition
Author
lazvertiigo
ID
333670
Card Set
Vector Spaces
Description
Vector Spaces
Updated