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Math Help - Subtraction in vector spaces

  1. #1
    Senior Member I-Think's Avatar
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    Subtraction in vector spaces

    We have to prove some of the basic properties of operations in vector spaces, and sometimes it's easy to make assumptions when doing them, so I just want to make sure I'm correct.

    Let V be a vector over a field F
    Prove (\lambda-\mu)x=\lambda{x}-\mu{x}

    Begin with
    (\lambda+\mu)x=\lambda{x}+\mu{x}
    Add the additive inverse
    (-1)\mu{x} to both sides
    (\lambda+\mu)x+(-1\mu{x})=\lambda{x}+\mu{x}+(-1\mu{x})

    Manipulating the L.H.S.
    (\lambda+\mu)x+(-\mu{x})=\lambda{x}+\mu{x}+(-\mu{x})<br />
    Employing associativity and commutativity
    \lambda{x}+\mu{x}+(-1\mu{x})=\mu{x}+(\lambda{x}+(-1\mu{x})<br />
    So
    \mu{x}+(\lambda{x}+(-1\mu{x})=\lambda{x}+\mu{x}+(-1\mu{x})<br />
    Add additive inverse -\mu{x} to both sides again and desired result is obtained.

    Is this correct?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by I-Think View Post
    We have to prove some of the basic properties of operations in vector spaces, and sometimes it's easy to make assumptions when doing them, so I just want to make sure I'm correct.

    Let V be a vector over a field F
    Prove (\lambda-\mu)x=\lambda{x}-\mu{x}

    Begin with
    (\lambda+\mu)x=\lambda{x}+\mu{x}
    Add the additive inverse
    (-1)\mu{x} to both sides
    (\lambda+\mu)x+(-1\mu{x})=\lambda{x}+\mu{x}+(-1\mu{x})

    Manipulating the L.H.S.
    (\lambda+\mu)x+(-\mu{x})=\lambda{x}+\mu{x}+(-\mu{x})<br />
    Employing associativity and commutativity
    \lambda{x}+\mu{x}+(-1\mu{x})=\mu{x}+(\lambda{x}+(-1\mu{x})<br />
    So
    \mu{x}+(\lambda{x}+(-1\mu{x})=\lambda{x}+\mu{x}+(-1\mu{x})<br />
    Add additive inverse -\mu{x} to both sides again and desired result is obtained.

    Is this correct?
    Yes, it is! But why couldn't you just say that (\lambda-\mu)x=(\lambda+(-\mu))x=\lambda x+(-\mu )x=\lambda x-\mu x since you seemed to use all the axioms I did in your proof?
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