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Homework Help: Sub spaces

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data
    If U and V are subsets of R^n, then the set U+V is
    defined by

    U+V={x:x=u+v,u in U, and v in V} prove that U and V are subspaces of R^n
    then the set U+V is a subspace of R^n.
    I am just having trouble proving U+V is a subspace.

    2. Relevant equations

    To be a sub-space...
    1. it needs to contain the zero vector
    2. x+y is in W whenever x and y are in W.
    3. ax is in W whenever x is in W and a is any scalar.

    3. The attempt at a solution

    1. U and V both contain the zero vector, so their sum will also contain the zero vector.
    2. any u1 plus u2 should be in U because U is a subspace, and any v1+v2 should be in V becuse V is a subspace. So (u1+v1)+(u2+v2)=(u1+u2)+(v1+v2)=u+v.
    3. below is just the matrix u+v times a
    a(v1+u1)=av1+au1
    (v2+u2)=av2+au2
    (v3+u3)=av3+au3

    Because u is in U, and v is in V then au must be in U, and av must b in V,
    and u+v is in U+V. Therefore a(U+V)must be in U+V.

    Is this sufficient?
     
  2. jcsd
  3. Feb 8, 2010 #2

    Dick

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    Homework Helper

    Yes, it's sufficient. Though I'm having a little trouble understanding why you felt you needed u1, u2, u3 and v1, v2, v3 in part 3. a(u+v)=au+av. Isn't that enough?
     
  4. Feb 8, 2010 #3
    I did that because I am a linear algebra newb lol.

    Thanks for the help.
     
  5. Feb 8, 2010 #4

    Dick

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    That's a good reason! :)
     
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