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Math Help - Linear functionals

  1. #1
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    Linear functionals

    Let V be a finite dimensional vector space over field F and let v belong to V,v not equal to 0.Then there exists some f in V* such that f(v) is not equal to 0.
    (where V* is dual space of V)
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  2. #2
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    Quote Originally Posted by math.dj View Post
    Let V be a finite dimensional vector space over field F and let v belong to V,v not equal to 0.Then there exists some f in V* such that f(v) is not equal to 0.
    (where V* is dual space of V)
    There is a basis \{v, e_2,\ldots,e_n\} of V with v ( =e_1) as its first element. Every element x of V has a unique expression x = \sum_{i=1}^n\alpha_ie_i as a linear combination of the basis vectors. Define f(x) = \alpha_1.
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    Quote Originally Posted by math.dj View Post
    Let V be a finite dimensional vector space over field F and let v belong to V,v not equal to 0.Then there exists some f in V* such that f(v) is not equal to 0.
    (where V* is dual space of V)
    f is a mapping f: V \rightarrow F, isn't it? It is termed a linear functional if it's a linear mapping from V into F.

    If \{v_1,...,v_n \} is a basis of Vover F, let f_1, ...,f_n \in V^* be the linear functionals defined by Kronecker delta notation:

    f_i(v_j) = \delta_{ij} =\left \{\begin{matrix} 1, & \mbox{if } i=j \\ 0, & \mbox{if } i \ne j \end{matrix}\right.

    Then set {f_1,...,f_n} is a basis for V^*. Anf those linear functionals f_i are unique since they're defined on a basis of V.
    Last edited by Opalg; January 1st 2010 at 11:27 AM. Reason: Fixed LaTeX
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