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Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy:

$$ Av_1 = v_1 \\ Av_2 = 2v_2 \\ Av_3 = 3v_3 $$

Prove that $\{v_1, v_2, v_3\}$ is linearly independent. (Hint: Start by showing that $\{v_1, v_2\}$ must be linearly independent.) I know if $A$ is non-singular,it is easy. But if $A$ is singular, I have no idea to get that.

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  • $\begingroup$ Do you know about eigenvectors and eigenvalues? $\endgroup$ – EuYu Apr 12 '13 at 4:26
  • $\begingroup$ Yes,but i do not think this question requires this technique $\endgroup$ – Yunhui Shi Apr 12 '13 at 4:30
  • $\begingroup$ if matrix is singular,it means that it's columns or rows are also linear dependent,maybe you should use this fact $\endgroup$ – dato datuashvili Apr 12 '13 at 5:01
  • $\begingroup$ The eigenvectors corresponding to distinct eigenvalues are always linearly independent $\endgroup$ – Shahab Apr 12 '13 at 5:36
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HINTS

  1. Suppose $v_1$ and $v_2$ are linearly dependent, so that $v_1 = cv_2$.
  2. What is $Av_1$? What about $A cv_2$? What about $cAv_2$?
  3. Do the same idea for $v_3$ independence.
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