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Homework Help: Subspaces homework help

  1. Jan 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the following are subspaces of R^m :

    (a) The set of all linear combinations of the vectors (1,0,1,0) and (0,1,0,1) (of R^4)
    (b) The set of all vectors of the form (a,b,a-b,a+b) of R^4

    2. Relevant equations

    3. The attempt at a solution

    (a) If (1,0,1,0) and (0,1,0,1) span R^4 , they are subspaces.

    (b) {a(1,0,1,1) + b(0,1,-1,1) | a,b [tex]\in[/tex]R}
    = span {(1,0,1,1) , (0,1,-1,1)}

    Are my answers incomplete ?
  2. jcsd
  3. Jan 21, 2009 #2
    Re: Subspaces

    To show its a subspace you need to show it is closed under the operation. What sort of space are we talking about here? Vector spaces?
  4. Jan 21, 2009 #3


    Staff: Mentor

    Re: Subspaces

    Two vectors can't possibly span R^4, a vector space of dimension 4. Also, the vectors themselves aren't subspaces. You are supposed to show that the set of all linear combinations of these two vectors is a subspace of R^4. To do that show that:
    1. The zero vector in R^4 is in this set (i.e., the set of linear combinations of the two vectors).
    2. Any two vectors in this set is also in the set.
    3. Any scalar multiple of a vector in this set is also in this set.
    See above.
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