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554-HW-13q

# 554-HW-13q - 13 Let V = C2 with the standard inner product...

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13) Let V = C 2 with the standard inner product. Let T be the linear operator defined by T 1 = (1 , - 2) , T 2 = ( i, - 1). Let α = ( x 1 , x 2 ) and find T * α. 13) Let T be the linear operator on C 2 defined by T 1 = (1 + i, 2) and T 2 = ( i, i ). Find the matrix of T * in the standard ordered basis. Does T commute with T * ? 13) Show that the range of T * is the orthogonal complement of null T , i.e. show R = R ( T * ) = (null( T )) = N . 13) Let V be a finite dimensional inner product space (fin dim IPS), and T a linear operator on V . If T is invertible, show that T * is invertible and that ( T * ) - 1 = ( T - 1 ) * . 13) Show that the product of 2 self-adjoint operators is self-adjoint iff the two operators commute. 13) Let V be a fin dim IPS over C . Let E be a projection operator / an idempotent operator on V . Prove E is self-adjoint iff E is normal, i.e. E = E * iff E * E = EE * . 13) Let V be a fin dim IPS over C . Let T be a linear operator on V . Show that T is self-adjoint iff ( Tx | x ) is real for all
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