TITLE: Proof of "Eigenvectors corresponding to different eigenvalues are linearly independent."
QUESTION [9 upvotes]: "Eigenvectors corresponding to different eigenvalues are linearly independent."
My professor told us this during a lecture, but gave no proof or explanation.

REPLY [11 votes]: Suppose that $Av=\lambda v$ and $Aw=\mu w$, $v,w\not=0$.
Assume $v,w$ are linearly dependent, then $v=c\cdot w$ for some scalar $c\not=0$. Then
$$\lambda v=Av=cAw=c \mu w = \mu v$$
That is, $\lambda=\mu$.