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Vector spacesHelp

  1. Nov 5, 2012 #1
    Hi i have two questions regarding this definition: "A vector space is a set that is closed under finite vector addition and scalar multiplication"

    First of all, is it correctly that a vector space simply is a set of rules that are assigned to a set of vector, the rules are addition and multiplication, and if we apply these rules the set of vecturs are in the vectorspace V?

    What does it mean that a set is closed under vector addition and scalar multiplication?
     
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  3. Nov 5, 2012 #2

    tiny-tim

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    hi christian0710! :smile:
    i don't understand :redface:
    it means the obvious … if you add two vectors (or multiply a vector by a scalar), the result is still a vector in the space
     
  4. Nov 5, 2012 #3
    Start with an arbitrary set of elements, call it V. Then define a special kind of function known as a binary operation on V which we call addition,
    ⊕ : V × V → V | (v,w) ↦ ⊕(v,w) = v ⊕ w.
    This map must satisfy certain obvious & intuitive axioms. Second define a second function known as scalar multiplication,
    ⊗ : F × V → V | (λ,w) ↦ ⊗(λ,w) = λ ⊗ w,
    where F is defined to be a field. This map must also satisfy certain axioms. Thus we have constructed the structure (V,⊕,⊗) known as a vector space. Only within a structure like this can we use the word vector meaningfully, i.e. a vector is an element of the set V in (V,⊕,⊗) that we can manipulate using the maps ⊕ & ⊗ (so the use of the word vector is an informal way of saying that the thing, say v, we're dealing with behaves in a certain manner, i.e. it behaves in the way our axioms allow). The way you've said it is a bit tautological, you can't really speak of a set of vectors until after you've constructed a vector space...

    Now as for closure, a function f : S → T is closed if f(x) ∈ T is always true. So for example the function + : {1,2} × {1,2} → {1,2} (which you can think of as a restriction of the addition function on the integers to {1,2} into {1,2}) is not closed on {1,2} because 2 + 2 = 4 ∉ {1,2}. So in the case of vector spaces you'd say that S is closed under vector addition and scalar multiplication if you can construct (S,⊕,⊗), where ⊕ & ⊗ are defined on S satisfying the vector space axioms & the operations are closed on S, basically ⊕(v,w) ∈ S & ⊗(λ,w) ∈ S always holds. This is useful because when you have a vector space (V,⊕,⊗) & you take some arbitrary subset of V, say W, you want to know whether W forms part of a vector space structure on it's own, i.e. (W,⊕,⊗) with ⊕ & ⊗ defined as they were on V (formally you take restrictions of these maps to the set W & want to know whether closure holds, i.e. (W,⊕|ᵂ,⊗|ᵂ), but you don't need to worry about this kind of formality too much).
     
  5. Nov 6, 2012 #4

    Fredrik

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    I agree with all of this. A vector space is a triple (V,⊕,⊗) that satisfies a number of conditions. The set V is called the underlying set of the vector space (V,⊕,⊗). A vector is a member of the underlying set of a vector space.

    The notation f : S → T means that T is the codomain of f. If T is the codomain of f, then it's always true that f(x)∈T for all x in S. The range f(X)={f(x)|x∈S} is always a subset of the codomain.

    The restriction of the addition operation on the integers to {1,2} is a function from {1,2} into the set of integers. Restriction only changes the domain, not the codomain.

    What you should be saying here is that the set {1,2} isn't closed under addition, because 1+2=3 isn't in {1,2}.
     
  6. Nov 6, 2012 #5

    Fredrik

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    I wouldn't say that a vector space is a "set of rules".

    Suppose that V is a set, ##\mathbb F## is a field (in pretty much all the interesting examples, ##\mathbb F## is either ℝ or ℂ), A is a map from V×V into V, and S is a map from ##\mathbb F##×V into V. The triple (V,A,S) is said to be a vector space if the eight conditions listed here are satisfied. The conditions are written out using the notations A(x,y)=x+y and S(a,x)=ax.

    If (V,A,S) is a vector space,
    • the map A is called addition, and we use the notation x+y instead of A(x,y).
    • the map S is called scalar multiplication, and we use the notation ax instead of S(a,x).
    • the set V is called the underlying set of the vector space (V,A,S).
    • the members of V are called vectors.
    • the members of ##\mathbb F## are called scalars.
    Addition is a function ##V\times V\to V##. Scalar multiplication is a function ##\mathbb F\times V\to V##, where ##\mathbb F## is a field. A set ##S\subset V## is said to be closed under addition if x+y is in S for all x,y in S. A set ##S\subset V## is said to be closed under scalar multiplication if λx is in S for all λ in ##\mathbb F## and all x in S.
     
  7. Nov 6, 2012 #6
    Thank you guys!! I'm on the way out, but will be studying and reading it this weekend. I appreciate your help in advance!
     
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