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Math Help - How to prove there is a unique linear map V->W for elements of the basis of V

  1. #1
    LHS
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    How to prove there is a unique linear map V->W for elements of the basis of V

    If I have a basis of V, say v1,..,vn and w1,...,wn elements of W, how can I show there is a unique transformation T(vi)=wi for all i 1..n?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint

    For all x\in V, x can be expressed univocally in the form x=\lambda_1v_1+\ldots+\lambda_nv_n\;(\lambda_i\in\  mathbb{K}).
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    LHS
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    So can I let T(x)= lambda[1]*w[1] + ... + lambda[n]*w[n], for some x element of V, then say the right hand side is uniquely determined due to lambda[i] being uniquely determined and as you can write x in terms of lambda[i]*v[i] there has to be a unique map between each v[i] to w[i]?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by LHS View Post
    So can I let T(x)= lambda[1]*w[1] + ... + lambda[n]*w[n], for some x element of V, then say the right hand side is uniquely determined due to lambda[i] being uniquely determined and as you can write x in terms of lambda[i]*v[i] there has to be a unique map between each v[i] to w[i]?

    Right, that proves the unicity of T. Besides, T(x)=\sum_{i=1}^{n}\lambda_i w_i is a linear map as a consequence of well known properties about coordinates.
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  5. #5
    LHS
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    Ah yes, so you can show it's a linear map, then let x=v[i] and you can show that T(v[i])= 0*w[1]+...+1*w[i]+...+0*w[n], then go about showing T is unique?
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