# Linearly independent vectors

• May 8th 2011, 04:54 PM
Oiler
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..
• May 8th 2011, 05:11 PM
alexmahone
Quote:

Originally Posted by Oiler
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.