Math 603-601 - Fall 2002

Assignment 6

1. Let T(x) = max(1 - 3|x|,0) be a tent function. Use the inner product < f,g > = ₀S ¹ f(x)g(x)dx to find the Gram matrix for the set B = {T(x), T(x - 1/3), T(x - 2/3), T(x - 1)}.

2. Find the Fourier series for the 2 periodic extension of the function f(x) = x defined on [-. ].

3. Use the Fourier series we found for the 2 periodic extension of |x| (see the summary for October 8) along with Parseval's equation to find the sum of the series
   1 + 3^-4 + 5^-4 + 7^-4 + 9^-4 + ...
   

4. Fix a vector a in R³. Find the matrix associated with the linear transformation L: R³ -> R³ defined via the cross product, L[x} = a×x. Use the standard basis, e₁ = i, e₂ = j, and e₃ = k

5. Let V = W = P₂ have the common basis B = D = {1,x,x²}. Suppose that L:P₂ -> P₂ is a linear transformation given by
   L[p] = (1 - x²)p'' - 2xp' + 3p,
   Find the matrix A that represents L. In addition, find the p for which L[p] = x² -x +1.

6. Find the eigenvalues and eigenvectors of the linear transformation L from the previous problem.

7. Find the solution to the system dx/dt = Ax, if x(0) = [1 -1]^T and A =

   3 4
   1 3