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Math Help - Linearly independent vectors

  1. #1
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    Linearly independent vectors

    Hey all,

    This problem was asked and I did not know how to solve it..
    Prove that if \lambda_1 and \lambda_2 are real and distinct eigenvalues for some matrix A, with corresponding eigenvectors V_1 and V_2, then V_1 and V_2 are linearly independent vectors.
    Thanks for the help..
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by Oiler View Post
    Hey all,

    This problem was asked and I did not know how to solve it..
    Prove that if \lambda_1 and \lambda_2 are real and distinct eigenvalues for some matrix A, with corresponding eigenvectors V_1 and V_2, then V_1 and V_2 are linearly independent vectors.
    Thanks for the help..
    Assume that v_1 and v_2 are linearly dependent. Then, v_2=kv_1

    Av_1=\lambda_1v_1

    Av_2=\lambda_2v_2

    \Rightarrow Akv_1=\lambda_2kv_1

    \Rightarrow Av_1=\lambda_2v_1

    \Rightarrow \lambda_1=\lambda_2 --- Contradiction because the problem states that \lambda_1 and \lambda_2 are distinct.

    Therefore, v_1 and v_2 are linearly independent.
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