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Math Help - Question about hyperplane

  1. #1
    Junior Member
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    Question about hyperplane

    Let V be a vector space over F field and f: V \rightarrow \mathbb{F} is nonzero linear functional. So f(\alpha u+\beta v) = \alpha f(u) + \beta f(v) for every u,v \in V and \alpha , \beta \in \mathbb{F}. Show that for every \lambda \in \mathbb{F} H_\lambda =\{ x \in V  |  f(x) = \lambda \} is a hyperplane.

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  2. #2
    MHF Contributor

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    Re: Question about hyperplane

    It's fairly straight forward isn't it? I would first look at H_0= \{x\in V| f(x)= 0\}. It's easy to show that H_0 is a subspace. Now, given any other \lambda Let y be a single vector such that f(y)= \lambda and show that for any x in H_\lambda, x- y is in H_0.
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