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Math Help - Lin. dep.

  1. #1
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    Lin. dep.

    Let f:X\to Y be a linear transformation. If x_1,\cdots , x_n are lin. dep., show that f(x_1),\cdots , f(x_n) are lin. dep.

    Since x_1,\cdots , x_n are lin. dep., there exist an x_i=\sum_{j\neq i}\lambda_j x_j where i=1,\cdots ,n.

    f(x_i)=f\left(\sum_{j\neq i}\lambda_j x_j\right)=\sum_{j\neq i}\lambda_j f(x_j)

    f(x_i)=\sum_{j\neq i}\lambda_j f(x_j)\Rightarrow 0=\left[\sum_{j\neq i}\lambda_j f(x_j)\right]-f(x_i)

    Which is a nontrivial sol. since not all \lambda_i are equal to zero.

    Correct?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Lin. dep.

    Quote Originally Posted by dwsmith View Post
    Correct?
    Yes, it is correct.
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