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Math Help - Linear Independence

  1. #1
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    Linear Independence

    Let V be a vector space over field F.Let V* be dual of V.If {f1,f2,...fk}subset of V* and {v1,v2,...vk}subset of V such that fi(vj)= 1 ,if i=j and 0 ,if i not equals j ..then prove that {f1,...fk} is linearly independent in V* and {v1,...vk} is linearly independent in V..

    i can show that {f1,...fk} is Linearly independent..i need help as to how to show {v1,...vk}are linearly independent in V..

    Thank you in advance
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  2. #2
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    Quote Originally Posted by math.dj View Post
    Let V be a vector space over field F.Let V* be dual of V.If {f1,f2,...fk}subset of V* and {v1,v2,...vk}subset of V such that fi(vj)= 1 ,if i=j and 0 ,if i not equals j ..then prove that {f1,...fk} is linearly independent in V* and {v1,...vk} is linearly independent in V..

    i can show that {f1,...fk} is Linearly independent..i need help as to how to show {v1,...vk}are linearly independent in V..

    Thank you in advance

    Suppose \sum\limits_{i=1}^ka_iv_i=0\,,\,\,a_i\in \mathbb{F} \Longrightarrow\,\forall\,1\leq j\leq k\,,\,\, 0=f_j(0) =f_j\left(\sum\limits_{i=1}^ka_iv_i\right)=\sum\li  mits_{i=1}^kf_j(a_iv_i)=\sum\limits_{i=1}^ka_if_j(  v_i)=a_j \Longrightarrow\,a_j=0\,\,\forall j and we're done.

    Tonio
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  3. #3
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    Thank you so much..
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