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Linear combination of linear combinations?

  1. Jan 25, 2013 #1
    When the book says "Members of [] are linear combinations of linear combinations of members of S". basically means the span of the members in subspace S. Since
    = {c1s1 +... + cnsn|c1...cnεR and s1...snεS} what does [] mean? does it mean a linear combination of atleast one linear or more linear combinations from ?
     
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  3. Jan 26, 2013 #2

    Fredrik

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    Let V be a finite-dimensional vector space. For all subsets S⊂V, is the set of all v in V such that v is equal to a linear combination of members of S. Since that makes a subset of V, the definition applies to as well. So [] is the set of all v in V such that v is equal to a linear combination of members of .

    It's possible to prove that if E,F⊂V, the following statements are equivalent (i.e. they're either all true or all false).

    (a) E is the set of all v in V such that v is a linear combination of members of F.
    (b) E is the intersection of all subspaces that have F as a subset.
    (c) E is the smallest subspace that has F as a subset. (This means that if E' is a subspace that has F as a subset, E⊂E').

    A set E for which these statements are true is, in your notation, denoted by [F]. If you only look at (a), it's not obvious that []=, but if you look at the other two statements, it is.

    Two alternative notations for : span S, ##\bigvee S##.
     
  4. Jan 26, 2013 #3
    Yeah I forgot to mention []=. But the book states that [] is a linear combination of linear combinations of the members of . Does this let's say set R={c1,1s1+...+cn,1sn,...,c1,m s1+...+cn,m sn} which is the set of all linear combinations of . And [] just means taking a linear combination of those members in R such as
    r1(c1,1s1+...+cnsn)+...+rn(c1,m s1+...+c n,m sn) ? Oh s1,...sn. are elements of S and S is a subspace of V
     
  5. Jan 26, 2013 #4

    Fredrik

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    I thought that was what you were asking about, not something you already knew.


    You mean "of S", right? (Not "of ").


    What does that mean? And did you again mean S when you wrote ?


    I don't understand what you're asking, but the meaning of [] is given by the definition of the [] notation.

    Maybe you meant to ask this: If we define R=, does that make []=[R]? The answer is of course yes. The [[]] notation isn't something entirely different from the [] notation. It's just the [] operation done twice.

    It doesn't have to be.

    I recommend that you start using LaTeX, or at least vBulletin's sup and sub tags. (Like this: E=mc2).
     
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