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
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