Let $v_1$ be an eigenvector of an $n$ x $n$ matrix $A$ corresponding to $\lambda_1$, and let $v_2, v_3$ be two linearly independent eigenvectors of $A$ corresponding to $\lambda_2$. Assume $\lambda_1\ne\lambda_2$. Show that $v_1, v_2, v_3$ are linearly independent.

I know this should be pretty easy, but I'm just not getting it...


Hint: Suppose $v_1=c_2v_2+c_3v_3$. Now multiply both sides by $A$.

  • 1
    $\begingroup$ @Jared $\lambda_2(c_2v_2+c_3v_3)=\lambda_2v_1$ by the assumption and also $\lambda_2(c_2v_2+c_3v_3)=\lambda_1v_1$ because of the matrix multiplication... $\endgroup$
    – Simon
    Mar 28 '16 at 2:44

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