We know that ifSis symmetric then eigenvectors associated with distinct eigenvalues are orthogonal. The proof

We know that if S is symmetric then eigenvectors associated with distinct eigenvaluesare orthogonal. The proof we discussed in class used the dot products 115(801 —)\101) =0 and 2135112 — A2112) = 0, and some simple algebraic manipulation. (a) Suppose first that A1 = 0 and A2 7E 0. Then U1 is in the null space of S and ’02is in the column space. Explain why this implies v1 _L’l)2. (b) Deduce the general case from part (a), by considering S — A1].