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Math Help - Hyperplane

  1. #1
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    Hyperplane

    Dear Colleagues,

    Could you please help me in solving the following problem:
    If Y is a subspace of a vector space X and codim Y=1, then every element of X/Y is called a hyperplane parallel to Y. Show that for any linear functional f\neq 0 on X, the set H_{1}=\{x\in X|f(x)=1\} is a hyperplane parallel to the null space N(f) of f.


    Regards,

    Raed.
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  2. #2
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help me in solving the following problem:
    If Y is a subspace of a vector space X and codim Y=1, then every element of X/Y is called a hyperplane parallel to Y. Show that for any linear functional f\neq 0 on X, the set H_{1}=\{x\in X|f(x)=1\} is a hyperplane parallel to the null space N(f) of f.


    Regards,

    Raed.

    Hints:

    S\Definition: A subspace H of a vector space V is called a hyperplane iff it is a maximal

    subspace of it, i.e. <H,v>=V\,,\,\,\forall v\in V

    Claim: a subspace of a vec. sp. is a hyperplane iff it is the kernel of a non-zero linear

    functional on the space

    Claim: a subspace of a finite dimensional v.s. is a hyperplane iff its dimension is one less

    that then space's.

    Tonio
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