3. Suppose \( U \) is a finite-dimensional real vector space and \( T \in \mathcal{L}(U) \) such that \( U \) has a basis consisting of eigenvectors of \( T \). Show that there exists an inner product on \( U \) that makes \( T \) into a self-adjoint operator. (3 Marks)
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Since \( U \) has a basis consisting of eigenvectors of \( T \), let this basis be \(\{v_1, v_2, \ldots, v_n\}\) where \( T(v_i) = \lambda_i v_i \) for each \( i \). Show more…
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