Let A be a given square matrix.
(a) Explain in detail why any nonzero scalar multiple of an eigenvector of A is also an eigenvector.
(b) Show that any nonzero linear combination of two eigenvectors v, w corresponding to the same eigenvalue is also an eigenvector.
(c) Prove that a linear combination cv + dw, with c, d ≠ 0, of two eigenvectors corresponding to different eigenvalues is never an eigenvector.