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Solution space of linear homogeneous PDE forms a vector space?

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Claim:
    The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space".

    Proof:
    Assume Lu=0 and Lv=0 (i.e. have two solutions)
    (i) By linearity, L(u+v)=Lu+Lv=0
    (ii) By linearity, L(au)=a(Lu)=(a)(0)=0
    => any linear combination of the solutions of a linear homoegenous PDE solves the PDE
    => it forms a vector space


    2. Relevant equations
    N/A

    3. The attempt at a solution and comments
    Now, I don't understand why ONLY by proving (i) and (ii) alone would lead us to conclude that it is a vector space. There are like TEN properties that we have to prove before we can say that it is a vector sapce, am I not right?
    Are there any theorem or alternative definition that they have been using?

    Thanks!
     
    Last edited: Sep 14, 2009
  2. jcsd
  3. Sep 14, 2009 #2

    lanedance

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    Homework Helper

    maybe consider the other axioms and test them, some may have been considered more bovious, but always worth checking
     
  4. Sep 14, 2009 #3

    Office_Shredder

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    Staff Emeritus
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    Gold Member

    Other things are satisfied by the definition of functions, for example u+v=v+u is obvious and not a property of being a solution of a PDE. It may be implicitly assumed that you're talking about the subset of the vector space of all functions from U to V (where U and V are Rm and Rn such that you'd be looking for solutions with that domain and codomain). In that case you only have to prove the two subspace properties
     
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