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Unformatted text preview: 1 1 1 1 1 1-1-1 1-1 1-1 1-1-1 1 . 4. (Bonus Question) (a) Prove or give a counterexample: the product of any two self-adjoint operators on an inner-product space is self-adjoint. (b) If T is a normal operator on a nite-dimensional complex inner product space such that T 9 = T 8 prove that T is self-adjoint and T 2 = T . 5. (Bonus Question) (a) Let T be a linear operator on a vector space V such that dim(Im( T )) = n . Prove that T has at most n + 1 distinct eigenvalues. (b) If T is a linear operator on a vector space V such that any vector of V is an eigenvector of T prove that T is a scalar multiple of the identity operator....
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- Winter '06