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Algebra - Proving a subspace

  1. Aug 30, 2011 #1
    1. The problem statement, all variables and given/known data
    Let U and W be subspaces of a vector space V
    Show that the set U + W = {v ∈ V : v = u + w, where u ∈ U and w ∈ W} is a subspace of V

    2. Relevant equations


    3. The attempt at a solution
    I understand from this that u and w are both vectors in a vector space V and that u+w is a vector in a vector space V but I'm not sure how to apply the rules to this problem in order to show that U + W is a subspace of V
     
  2. jcsd
  3. Aug 30, 2011 #2

    HallsofIvy

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    Remember that the definition of "subspace" is a subset that is closed under vector addition and scalar multiplication. If x is in U+ W, then there exist vectors u in U and v in V such that x= u+ v. Similarly, if y is in U+ W, then there exist vectors u' in U and v' in V such that y= u'+ v'. Can you say the same thing about x+ y? x+ y is equal to the sum of what vectors in U and W? What about ax where a is a scalar?
     
  4. Aug 30, 2011 #3
    Well if x and y are vectors in U + W then x+y will be a vector in U + W. x + y = u + w + u' + w' = (u + u') + (w + w')? u + u' is in U and w + w' is in W. Therefore x + y is in U + W. So U + W is closed under addition. Is this right?
     
    Last edited: Aug 30, 2011
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