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Math Help - Codimension of the null space

  1. #1
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    Codimension of the null space

    Dear Colleagues,

    Could you please help me in the following problem:

    Let f\neq0 be any linear functional on a vector space X show that in the quotient space X/N(f) the codim N(f)=1. Here N(f) denotes the null space of f, and codim means the dimension of X/N(f).

    Remark: two elements x_{1}, x_{2}\in X belong to the same element of the quotient space X/N(f) if and only if f(x_{1})=f(x_{2}).


    Regards,

    Raed.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint :


    If f\neq 0 then, \textrm{Im}f=\mathbb{K} i.e. \dim (\textrm{Im}f)=1.
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  3. #3
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    I do not understand.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by raed View Post
    I do not understand.

    What does mean f:X \to \mathbb{K} is different from 0 ?
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  5. #5
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    Quote Originally Posted by raed View Post
    I do not understand.
    Please be more specific. What part do you not understand?
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  6. #6
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    Do you understand what a linear functional is?

    (I puzzled over FernandoRevilla's response until I realized I had read "functional" but was still thinking "transformation"!)
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