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V is a finite dimensional real inner product space, T is an operator on V. T is called adjoint if T* = -T. Establish the following results about skew adjoint operators, <, >. a) Let U be an eigenvector for T² and -a, with a > 0 an eigenvalue. Set S = Sp{e,T,}. Show S is T-inyariant and there exists an O.N. basis of S such that T restricted to U has matrix for some k > 0. k b) For T skew adjoint and non-zero show there exists an O.N. basis B such that the matrix representing T with respect to B has the form: Os O's M O's Os Os O's O's M where each M is a 2x2 matrix of the form with k c) For a > 0 diagonalize over the complex numbers. d) For Z a fixed vector in R°(==00 define T by T(u) = zou (the cross product). Establish that T is skew adjoint. Find the matrix which represents T with respect to the standard basis. -k Find an O.N. basis B such that the matrix representing T has the form k 0 0 0 0 0

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