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MBA IT, Mater in Science and Technology
Devry
Jul-1996 - Jul-2000
Professor
Devry University
Mar-2010 - Oct-2016
problem 3 and 4 in the attached file please give detailed procedures thank you
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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(+ )
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