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Math Help - Null Space

  1. #1
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    Null Space

    Please can you guys help me to solve the following questions

    Q1.if Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of suitable linear functional f on X, which is uniquely determined to within a scalar multiple.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kinkong View Post
    Please can you guys help me to solve the following questions

    Q1.if Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of suitable linear functional f on X, which is uniquely determined to within a scalar multiple.
    Use the exact same logic as my other post. Take a basis \{x_1,\cdots,x_{n-1}\} for Z and extend it to a basis \{x_1,\cdots,x_n\} for X and define \varphi:X\to F by \varphi(x_k)=\delta_{k,n}. To prove it's unique up to a scaling factor note that any linear functional \psi must have the same form except \psi(x_n)\ne 0 may be different and so evidently \varphi=\frac{1}{\psi(x_n)}\psi
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  3. #3
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    note that the linear functional φ in this case is the dual vector to the basis element xn.

    note also that since n = dim(X) = null(φ) + rank(φ) = null(φ) + 1, dim(null(φ)) = n-1.

    it is a theorem that every hyper-space (or hyper-plane) arises in this fashion.
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