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Vector Space Sums

  1. Aug 13, 2008 #1
    The sum of two subspaces seems a simple enough concept to me, but I must be misunderstanding it since I don't understand why Axler gives an answer he does in Linear Algebra Done Right.

    Suppose U and W are subspaces of some vector space V.

    [tex]U = \{(x, 0, 0) \in \textbf{F}^3 : x \in \textbf{F}\} \text{ and } W = \{(y, y, 0) \in \textbf{F}^3 : y \in \textbf{F}\}.[/tex]

    The sum is given as follows:

    [tex]U + W = \{(x, y, 0) : x, y \in \textbf{F}\}.[/tex]

    Whereas it seems to me it should be:

    [tex]U + W = \{(x+y, y, 0) : x, y \in \textbf{F}\}.[/tex]

    Am I misunderstanding the concept of sum, and it does not really mean that all the elements in [tex]U + W[/tex] should have the form [tex](x, 0, 0) + (y, y, 0)[/tex], or [tex](x+y, y, 0)[/tex]?
  2. jcsd
  3. Aug 13, 2008 #2


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    since x can be anything, it can be anything -y, so the set of vectors of form (x,y) where x and y bare anything, is the same as the set of vectors (x+y,y) where x and y are anything.

    so you are also right!
  4. Aug 13, 2008 #3
    Thanks so much. That makes sense. I didn't think about simplifying x + y.
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