Let A be a given square matrix. (a) Explain in detail why any nonzero scalar multiple of
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(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.
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