TITLE: Eigenvectors are linearly independent?
QUESTION [14 upvotes]: Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Could someone give me a geometric interpretation of the theorem? Thanks!

REPLY [9 votes]: $Av_1=\lambda_1v_1$, $Av_2=\lambda_2v_2$, $\lambda_1\neq\lambda_2$, $v_1,v_2\neq0$. Suppose $v_2=cv_1$. Then $Av_2=\lambda_2v_2=c\lambda_2v_1$ and $Av_2=Acv_1=cAv_1=c\lambda_1v_1$, hence $c(\lambda_2-\lambda_1)=0\implies c=0\implies v_2=0$.