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Math Help - Linear Functionals

  1. #1
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    Linear Functionals

    Dear Colleagues,

    Could you please help me in solving the following problem:
    If f_{1}, ....,f_{p} are linear functionals on an n-dimensional vector space X, where p<n, show that there is a vector space x\neq 0 in X such that f_{1}(x)=0,..., f_{p}(x)=0. What consequences does this result have with respect to linear equations.

    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 following problem:
    If f_{1}, ....,f_{p} are linear functionals on an n-dimensional vector space X, where p<n, show that there is a vector space x\neq 0 in X such that f_{1}(x)=0,..., f_{p}(x)=0. What consequences does this result have with respect to linear equations.

    Regards,

    Raed.
    Merely note that since p<n you have that \text{span}\left\{f_1,\cdots,f_p\right\}\overset{\  text{de}\text{f.}}{=}V is such that \dim_F V<\dim_F \text{Hom}\left(X,F\right)=n. Thus, it follows that \dim\text{Ann }V=n-\din_F V>0 and so in particular there exists some non-zero \Phi\in\text{Ann }V which by definition must satisfy \Phi(f_k)=0 for k=1,\cdots,p. But it is a common fact that for finite dimensional vector spaces every element \Psi of V^{\ast\ast} is of the form of some evaluation functional f\mapsto f(x) for some fixed x\in V. Thus, in particular there exists some non-zero v\in V such that \Psi(f)=f(v) for every f\in \text{Hom}(V,F) and so in particular f_k(v)=\Phi(f)=0 for k=1,\cdots,p.
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    Merely note that since p<n you have that \text{span}\left\{f_1,\cdots,f_p\right\}\overset{\  text{de}\text{f.}}{=}V is such that \dim_F V<\dim_F \text{Hom}\left(X,F\right)=n. Thus, it follows that \dim\text{Ann }V=n-\din_F V>0 and so in particular there exists some non-zero \Phi\in\text{Ann }V which by definition must satisfy \Phi(f_k)=0 for k=1,\cdots,p. But it is a common fact that for finite dimensional vector spaces every element \Psi of V^{\ast\ast} is of the form of some evaluation functional f\mapsto f(x) for some fixed x\in V. Thus, in particular there exists some non-zero v\in V such that \Psi(f)=f(v) for every f\in \text{Hom}(V,F) and so in particular f_k(v)=\Phi(f)=0 for k=1,\cdots,p.
    Thank you very much for your reply. But what do you mean by {Hom}(V,F) and {Ann }V.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by raed View Post
    Thank you very much for your reply. But what do you mean by {Hom}(V,F) and {Ann }V.
    Dual space and annihilator respectively.
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  5. #5
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    In fact I am not able to understand the post. Could you please resolve the problem in another way if it is possible.

    Regards.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by raed View Post
    In fact I am not able to understand the post. Could you please resolve the problem in another way if it is possible.

    Regards.
    No, I cannot. But you can.
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