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

    Suppose V= {(x, 0, 0)} in $\displaystyle R^3$ and W= {(0, y, 0)} in $\displaystyle R^3$. Then $\displaystyle U\cup V$ 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 $\displaystyle V\cup W$ is any element of $\displaystyle V$ or of $\displaystyle W$. however, an element of $\displaystyle V \oplus W$ is a vector in which you can write it as a sum of an element of $\displaystyle V$ and of $\displaystyle W$. indeed, $\displaystyle V \oplus W = \mbox{span }{V\cup W}$
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