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Math Help - Dimension, onto, kernels

  1. #1
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    Dimension, onto, kernels

    Let V be a finite dimensional vector space over a field F. Denote by n the dimension of V, and suppose that we are given a vector v \neq0 in V.
    Show that there exits a linear map T: V-->F such that T(v) \neq0.(We view F as a one dimensional vector space over itself).
    Show that any such T is onto, and that the kernel of T must have dimension n-1
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  2. #2
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    Quote Originally Posted by studentmath92 View Post
    Let V be a finite dimensional vector space over a field F. Denote by n the dimension of V, and suppose that we are given a vector v \neq0 in V.
    Show that there exits a linear map T: V-->F such that T(v) \neq0.(We view F as a one dimensional vector space over itself).
    Show that any such T is onto, and that the kernel of T must have dimension n-1
    Let v1,v2,....,vn be basis of V

    v = a1v1 + a2v2 + .......... anvn
    as v \neq0 not all ai's are = 0. Let a1 \neq0

    Define T as such
    T(v1) = 1
    T(vi) = 0 for all other vectors in the basis.

    This T satisfies all the properties you mentioned. You can check them one by one.
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