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Math Help - Difference between union of 2 vector spaces and the direct sum of them

  1. #1
    MHF Contributor arbolis's Avatar
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    Difference between union of 2 vector spaces and the direct sum of them

    What's the difference between V \cup W and V \oplus W, V and W are vector spaces. I didn't find the answer on the Internet.
    A friend told me that  V+W= \text {span} (V \cup W). But then if V and W are \mathbb{R}^2 as vector space over \mathbb{R} then I get that \mathbb{R}^2+\mathbb{R}^2 is formed by ... ahh well I'm unsure of everything.
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    Quote Originally Posted by arbolis View Post
    What's the difference between V \cup W and V \oplus W, V and W are vector spaces. I didn't find the answer on the Internet.
    A friend told me that  V+W= \text {span} (V \cup W). But then if V and W are \mathbb{R}^2 as vector space over \mathbb{R} then I get that \mathbb{R}^2+\mathbb{R}^2 is formed by ... ahh well I'm unsure of everything.
    Well, yes, R^2\cup R^2= R^2 and since R^2 is already a subspace its span is also just R^2. That's not a very good example!

    Suppose V= {(x, 0, 0)} in R^3 and W= {(0, y, 0)} in R^3. Then U\cup V is the set of all vectors of the form (x, 0, 0) or (0, y, 0). Geometrically, that is the x-axis and the y-axis but NOT the points in between. That is not a subspace because, while it contains (1, 0, 0) and (0, 1, 0), it does NOT contain their sum, (1, 1, 0). The direct sum of V and W is the xy-plane: {(x, y, 0)}} which includes (1, 0, 0), (0, 1, 0) and (1, 1, 0).
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  3. #3
    MHF Contributor kalagota's Avatar
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    in general, an element of V\cup W is any element of V or of W. however, an element of V \oplus W is a vector in which you can write it as a sum of an element of V and of W. indeed, V \oplus W = \mbox{span }{V\cup W}
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