Linearly independent vectors

• May 8th 2011, 03:54 PM
Oiler
Linearly independent vectors
Hey all,

This problem was asked and I did not know how to solve it..
Prove that if $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$ are real and distinct eigenvalues for some matrix A, with corresponding eigenvectors $\displaystyle V_1$ and $\displaystyle V_2$, then $\displaystyle V_1$ and $\displaystyle V_2$ are linearly independent vectors.
Thanks for the help..
• May 8th 2011, 04:11 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 $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$ are real and distinct eigenvalues for some matrix A, with corresponding eigenvectors $\displaystyle V_1$ and $\displaystyle V_2$, then $\displaystyle V_1$ and $\displaystyle V_2$ are linearly independent vectors.
Thanks for the help..

Assume that $\displaystyle v_1$ and $\displaystyle v_2$ are linearly dependent. Then, $\displaystyle v_2=kv_1$

$\displaystyle Av_1=\lambda_1v_1$

$\displaystyle Av_2=\lambda_2v_2$

$\displaystyle \Rightarrow Akv_1=\lambda_2kv_1$

$\displaystyle \Rightarrow Av_1=\lambda_2v_1$

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

Therefore, $\displaystyle v_1$ and $\displaystyle v_2$ are linearly independent.