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Math Help - Prove that span{x,y}=span{y,z}

  1. #1
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    Prove that span{x,y}=span{y,z}

    I have never liked span questions, so I could use some help with this.

    First, suppose that V is a vector space over a field F.

    Now suppose x,y,z\in V satisfies x+y+z=0. Prove that span\{x,y\}=span\{y,z\}.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Take x over to the other side, y+z=-x, to see that x \in Span(y, z) \Rightarrow Span(x, y) \subseteq Span(y, z) (can you see why?)

    Then, do it the other way!
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  3. #3
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    Saying that v is in the span of x and y means v= ax+ by for some scalars a and b.

    Since x+ y+ z= 0, x= -y- z so v= a(-y- z)+ by.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Saying that v is in the span of x and y means v= ax+ by for some scalars a and b.

    Since x+ y+ z= 0, x= -y- z so v= a(-y- z)+ by.
    This means that v=a(-y-z)+by implies that v\in span(y,z), right? Or am I misinterpreting something?
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  5. #5
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    Hi
    ..it also means,
    v=(b-a)y-az\Rightarrow v\in Span\left \{ y,z \right \}
    v,y and z are vectors.
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  6. #6
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    Quote Originally Posted by Runty View Post
    This means that v=a(-y-z)+by implies that v\in span(y,z), right? Or am I misinterpreting something?
    Well, v= a(-y- z)+ by= (-a+ b)y+ (-a)z so that v is a linear combination of y and z and therefore is in span(y,z).

    You will also need to prove the other way: if u is in span(y, z), then it is in span(x, y).
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