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Math Help - Question about inner products

  1. #1
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    Question about inner products

    Hi, I have a question I need to do that look quite axiom-based, but I've been working through it for a while and I just can't seem to get out the answer that I need!

    We assume that V is a real vector space, and {\left\langle {x,y} \right\rangle _1} and {\left\langle {x,y} \right\rangle _2} are inner products defined on V.

    Assuming that \forall x \in V,{\rm{ }}{\left\langle {x,x} \right\rangle _1} = {\left\langle {x,x} \right\rangle _2}, I want to show that \forall x,y \in V,{\rm{ }}{\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y} \right\rangle _2}.

    I've tried methods like {\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y + x - x} \right\rangle _1} = {\left\langle {x,y} \right\rangle _1} + {\left\langle {x,x} \right\rangle _1} - {\left\langle {x,x} \right\rangle _1}, but this doesn't seem to lead me anywhere. Can anyone help?

    Thanks!
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  2. #2
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    Quote Originally Posted by lalala23 View Post
    Hi, I have a question I need to do that look quite axiom-based, but I've been working through it for a while and I just can't seem to get out the answer that I need!

    We assume that V is a real vector space, and {\left\langle {x,y} \right\rangle _1} and {\left\langle {x,y} \right\rangle _2} are inner products defined on V.

    Assuming that \forall x \in V,{\rm{ }}{\left\langle {x,x} \right\rangle _1} = {\left\langle {x,x} \right\rangle _2}, I want to show that \forall x,y \in V,{\rm{ }}{\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y} \right\rangle _2}.

    I've tried methods like {\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y + x - x} \right\rangle _1} = {\left\langle {x,y} \right\rangle _1} + {\left\langle {x,x} \right\rangle _1} - {\left\langle {x,x} \right\rangle _1}, but this doesn't seem to lead me anywhere. Can anyone help?

    Thanks!

    Develop <x-y,x-y>_1=<x-y,x-y>_2...as simple as that.

    Tonio
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