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

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


    Regards,

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


    If $\displaystyle f\neq 0$ then, $\displaystyle \textrm{Im}f=\mathbb{K}$ i.e. $\displaystyle \dim (\textrm{Im}f)=1$.
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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 $\displaystyle f:X \to \mathbb{K}$ is different from $\displaystyle 0$ ?
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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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