1.) Show that any nonzero linear combination of two eigenvectors v,w corresponging to the same eigenvalue is also an eigenvector.
2.) Prove that a linear combination $cv+dw$, with $c,d \ne 0$, of two eigenvectors corresponding to different eigenvalues is never an eigenvector.
3.) Let $\lambda$ be a real eigenvalue of the real n x n matrix A, and $v_1,...,v_k$ a basis for the associated eigenspace $V_{\lambda}$. Suppose $w \in \mathbb{C^n}$ is a complex eigenvector, so $Aw = \lambda w$. Prove that $w = c_1v_1 + ... + c_kv_k$ is a complex linear combination of the real eigenspace basis.
For 1 and 2 I know that if two eigenvectors $\vec{v}_1$ and $\vec{v}_2$ are associated with the same eigenvalue then any linear combination of those two is also an eigenvector associated with that same eigenvalue. But, if two eigenvectors $\vec{v}_1$ and $\vec{v}_2$ are associated with different eigenvalues then the sum $\vec{v}_1+\vec{v}_2$ need not be related to the eigenvalue of either one. In fact, just the opposite. If the eigenvalues are different then the eigenvectors are not linearly related. But I can show this using a proof?
For 3. I am not too sure.