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Math Help - Null Space of A linear Functional

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    Null Space of A linear Functional

    Dear Colleagues,

    Could you please help me in solving the problem:
    If Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of a suitable linear functional f on X, which is uniquely determined to within a scalar multiple.

    Regards,

    Raed.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help me in solving the problem:
    If Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of a suitable linear functional f on X, which is uniquely determined to within a scalar multiple.

    Regards,

    Raed.
    Let Z have a basis \{x_1,\cdots,x_{n-1}\} and extend it to a basis \{x_1,\cdots,x_n\} for X. Then, merely define \varphi:X\to F by x_k\mapsto \delta_{k,n} and extend by linearity.


    Show then that \ker\varphi= Z and moreover that the only other way to construct a linear functional was to extend the basis for Z to a basis for X by picking some other x'_n\in \text{span}\{x_n\} which then amounts to any other such linear functional looking like x_k\to \alpha\delta_{k,n} where x'_n=\alpha x_n. etc.


    Now prove all of that
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help me in solving the problem:
    If Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of a suitable linear functional f on X, which is uniquely determined to within a scalar multiple.

    Regards,

    Raed.


    Let \{x_1,\ldots ,x_{n-1}\} be a basis for Z , and complete this to a basis \{x_1,...,x_{n-1},x_n\} of

    the whole n-dimensional space.

    Now define f:X\rightarrow \mathbb{F}\,,\,\,\mathbb{F}= the definition field, by

    f(x_i)=\left\{\begin{array}{ll}0&\mbox{ , if }i=1,...,n-1\\1&\mbox{ , if }i=n\end{array}\right. and extend the definition by linearity.

    Show now that  Z=\ker f

    Tonio
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    Thank you very much.
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  5. #5
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    How the extension by linearity can be done.

    Regards.
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    Quote Originally Posted by raed View Post
    How the extension by linearity can be done.

    Regards.

    This is a standar procedure: Just write any element of the space as a linear combination

    of the basis and define \displaystyle{f(v)=f\left(\sum\limits^n_{i=1}a_ix_  i\right):=\sum\limits^n_{i=1}a_if(x_i) ...

    Tonio
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    I undetrstand you. Thank you very much.

    Best Regards.
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