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Math Help - Linear functional problem

  1. #1
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    Linear functional problem

    I have been strugling with this problem for a while now, would appreciate any direction.

    If l\neq 0 is a linear functional on an n-dimensional vector space V.
    Show that there exists a basis of V B=\{v_{1},...v_{n}\}, such that for every v\inV, if v=\alpha_{1}v_{1}+...+\alpha_{n}v_{n}, then l(v)=\alpha_{1}.

    My only idea so far is that if I can show that there exists a basis of V B=\{v_{1},...v_{n}\} such that l(v_{1})=1 and l(v_{i})=0 for i=2,...,n then I will have what I need. I was thinking of chossing a v such that l(v)=1 and then completing that to an orthonormal basis, but I can seem to get anywhere with this.
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  2. #2
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    Quote Originally Posted by skyking View Post
    I have been strugling with this problem for a while now, would appreciate any direction.

    If l\neq 0 is a linear functional on an n-dimensional vector space V.
    Show that there exists a basis of V B=\{v_{1},...v_{n}\}, such that for every v\inV, if v=\alpha_{1}v_{1}+...+\alpha_{n}v_{n}, then l(v)=\alpha_{1}.

    My only idea so far is that if I can show that there exists a basis of V B=\{v_{1},...v_{n}\} such that l(v_{1})=1 and l(v_{i})=0 for i=2,...,n then I will have what I need. I was thinking of chossing a v such that l(v)=1 and then completing that to an orthonormal basis, but I can seem to get anywhere with this.

    Look at \ker l . This is a subspace of dimension n -1, so choose from here a base \{v_2,\ldots,v_n\} complete it to a base \{v_1,\ldots,v_n\}

    of the whole space, and since then l(v_1)=k\in\mathbb{F}\,,\,k\neq 0, instead v_1 take \frac{1}{k}v_1

    Tonio
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