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Math Help - prove or disprove: statement about Span

  1. #1
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    prove or disprove: statement about Span

    V is a linear space. A,B\in V are sets, and W is subspace:

    Span(A\cup B\cup W)=(Span(A)+Span(B))\cup W

    i just need help to figure out if it's true or false.

    thanks in advanced!
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  2. #2
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    Re: prove or disprove: statement about Span

    Do you know what "span" means? It looks to me like a straight forward case of applying the definition.

    What have you tried?
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    Re: prove or disprove: statement about Span

    suppose V is the vector space R3, A = {(0,0,1)} B = {(0,1,0)} and W is the x-axis (vectors of the form (x,0,0)). what is the LHS and RHS then?
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  4. #4
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    Re: prove or disprove: statement about Span

    Hi guys, and thank you.

    Denovo, that's the first example that crossed my mind, by this example the LHS and RHS are equal:
    Span\left \{ (0,1,0), (0,0,1), (x,0,0)|x\in \mathbb{R} \right \}=\left \{ (x,y,z)|x,y,z\in \mathbb{R}\right \}=\mathbb{R}^3
    is equal to:
    Span(A)=\left \{ (0,y,0)|y\in \mathbb{R}\right \}+Span(B)=\left \{ (0,0,z)|z\in \mathbb{R}\right \}\cup \left \{ (x,0,0)|x\in \mathbb{R}\right \}=\left \{ (x,y,z)|x,y,z\in \mathbb{R}\right \}=\mathbb{R}^3

    but my intuition misled me so many times, and i wasn't sure that proves it.
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  5. #5
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    Re: prove or disprove: statement about Span

    no, they are not equal. the point (3,4,5) is in Span(AUBUW), since:

    (3,4,5) = 4(0,1,0) + 5(0,0,1) + (3,0,0), and (0,1,0) is in A, so in AUBUW, (0,0,1) is in B, so in AUBUW, and (3,0,0) is in W, so is in AUBUW.

    is (3,4,5) in Span(AUB)UW? if so, it would have to be in either Span(AUB), or W, or both. is this true?
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  6. #6
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    Re: prove or disprove: statement about Span

    my bad.

    (3,4,5) is neither in (Span(A)+Span(B)) nor in W, and therefore is not in (Span(A)+Span(B))\cup W.

    thanks.
    this disproves it.

    this is all so confusing...
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  7. #7
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    Re: prove or disprove: statement about Span

    The best way to avoid confusion in mathematics is to learnthe definitions well!
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  8. #8
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    Re: prove or disprove: statement about Span

    and the moral of our story is: span and union don't play well together, and should therefore sleep in separate rooms.
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