Let $\displaystyle \lambda$ and $\displaystyle \mu$ be distinct eigenvalues for $\displaystyle T:R^n \rightarrow R^n$ and let the corresponding eigenspaces be $\displaystyle V_\lambda$ and $\displaystyle V_\mu$ respectively.

Prove that the only vector common to both $\displaystyle V_\lambda$ and $\displaystyle V_\mu$ is the zero vector.