(1).  (10 points total) Suppose that ! 1(1)# is an eigenvector of a matrix A corresponding to eigenvalue 3, and that ! 1(2)#    is an eigenvector of matrix A corresponding to eigenvalue − 2. Compute A!  !3(4)# .  Justify the steps in your solution.

(2). (15 points total) Let V be a two-dimensional linear vector space with an inner product. Let B1  = {| v1  >, | v2   >} be an orthonormal basis for V. Let  be a linear operator on V, which is represented in the basis  B1  by the matrix

A"  = ! i(2)    i#

(a). (5 points) Explain and justify how you know the operator   is diagonalizable.

(b). (10 points) Diagonalize the operator  by finding a new orthonormal basis B2  = {| e1  >, | e2  >} in which   is represented by a diagonal matrix A!. To receive full      credit your answer must include expressions for (i) the new basis vectors which comprise B2 defined in terms of the original basis vectors which comprise  B1 , (ii) the operator      defined in terms of the basis vectors in  B2 , (iii) the change of basis matrix, and (iv) the    matrix A!  representing the operator   in the new basis.

(3). (25 points total) Let V be a vector space. Consider a Hermitian operator  that is represented in an orthonormal basis {|e1   >, |e2   >} of  V by the matrix H  = ! i(0)    i#.

(a). (5 points) Find the eigenvalues and normalized eigenvectors of the matrix H.

(b).  (5 points) Express the corresponding normalized eigenvectors {|v1   >, |v2   >} of  in terms of the original orthonormal basis vectors {|e1   >, |e2   >}. Do the eigenvectors {|v1  >, |v2  >}  form a basis of V? If so, do they form an orthonormal basis? Justify your answer.

(c). (5 points) Use the results of part (a) to construct the Green’s Function operator  for the linear operator   in terms of the basis vectors  {|e1   >, |e2   >}.   [In part  (e) you will   use the Green’s Function to solve the equation |y  >=  |f  >, for |y  >, where |f > is a   (given) inhomogeneous term.]

(d). (5 points) Find the matrix G that represents the Green’s Function in the basis {|e1  >, |e2  >}.

(e). (5 points) Use the Green’s function to solve for |y  > in the equation:

|y >= |f >

where the inhomogeneous driving term is given by |f  > =  |e1   > +|e2   >.

(4). (25 points total) This problem involves the Sturm-Liouville system:

d zun (x) dxz

= λnun (x),                  0 ≤ x ≤ 1;

dun (0)             dun (1)

dx             ,        dx

(a). (8 points) Identify the Sturm Liouville operator  and write down the appropriate inner product and weight function that makes   Hermitian. Identify the eigenvalues and normalized eigenfunctions of  .

(b). (7 points) Compute the expansion of the function f(x) = x in terms of the eigenfunctions in part (a).

(c). (7 points) Use separation of variable to solve the heat equation for the temperature

profile T(x, t) of a one-dimensional rod, where the ends of the rod at x = 0 and x=1 are   insulated (i.e. no heat flows out of the ends), and the initial temperature profile is given in part (b). This situation is described by the partial differential equation, with boundary and initial conditions as specified below:

a：T(x,t) ax：

aT(x,t)

at    ,

0 ≤ x ≤ 1,  t ≥ 0

aTx(0),t) = 0,   aT,t) = 0, T(x, 0) = x

(d). (3 points) How does the solution behave as t → ∞ (i.e. what is lim T(x, t))? t→ &

(5).  (25 points total) In this problem you will find the steady state temperature on both   the inside and the outside a sphere of radius “ = ' when the surface of the sphere is held at a fixed temperature distribution u(r = 1,θ,φ).

(a). (5 points) The fixed boundary condition for the temperature on the surface the sphere is  u (r = 1, θ, φ) = cos(θ) − 3sin2 (θ). Expand this function as a Legendre series in      x = cosθ. For this expansion, you can use the Legendre Polynomials  r! (cosθ) , the

normalized Legendre Polynomials p! (cosθ) , or the m=0 spherical harmonics Y!,m=0(θ, φ) (whichever you prefer).

(b). (7 points) Solve for the steady state temperature distribution u(r, θ) inside the sphere 0 ≤ r ≤ 1 for the fixed boundary conditions specified in part (a). Note: by symmetry, the solution is independent of φ because the boundary condition does not depend on φ .

(c). (7 points) Solve for the steady state temperature distribution u(r, θ)  outside the sphere 1 ≤ r < ∞  for the fixed boundary conditions specified in part (a).

(d). (6 points) Combine the results from parts (b) and (c) to obtain the complete solution u (r, θ) everywhere in space. Then set θ  = 0 and obtain the radial solution u(r) for that specific angle. Sketch a graph of your result for u(r) for  0 < r < ¥  for the angle θ  = 0.