Chem 120A, Fall 2016 Problem Set 1 Due in G13 Gilman on Sep 2 at 4pm 1. Show that two of the following operators are linear and that one is not. (()) = + 3() (( )) = 5 1 () 2 (()) = ( )3 () 0 2. Consider the inner product space consisting of all linear combinations of sin() and cos() with the inner product: | = ( )( ) 0 a) What is the matrix representation of the operator if we take sin() and cos() to be our basis functions? b) Show that the vectors | = and | = are orthogonal in this space. c) Find the scalar so that | and | form an orthonormal basis. d) What is the matrix representation of in this basis? e) What is special about the basis formed by | and |? 3. Using the inner product | = ()( ) in a vector space for which all vectors satisfy () = () = 0, show that is a Hermitian operator but that is not. 4. Consider the vector space of all 20 by 20 complex-valued matrices. Show that one of the following forms involving the matrix trace (Tr) is a valid inner product but that the other is not. Here we use + to represent the conjugate transpose of the matrix . (Hint: singular value decompositions are very useful things, and matrix traces and unitary matrices have many interesting properties.) | = Tr() | = Tr(+ )