Math 311-501 — Current Assignment

Assignment 11 - Due Thursday, 26 April 2007

• Do the following problems.
  1. Section 3.5B (pgs. 137-138): 17

  2. Review problems, chapter 3 (p. 172): 22

  3. Let L : P₂ → P₂ be defined by L[p]= (x² - x + 3)p'' − (4x − 1)p' + 6p.
     1. Show that L is linear.
     2. Find the matrix A of L relative to the standard basis B = {1, x, x²}.
     3. Find bases for the null space and image (column space) of A.
     4. Use the bases you found above to write down bases for the null space and image of L.

  4. In the diagram for the spring-mass system in problem 6 of Test 2, use 4k (rather than 2k) for the middle spring's constant. Find the normal modes of this system, given that Newton's laws applied to the system give these equations of motion:
     
     

  5. In class we discussed the Gram matrix G for a least squares problem. Given a linearly independent set of vectors {w₁,..., w_n} and an inner product < , >, we define the entries of G via
     G_j,k = <w_j, w_k>.
     For the vectors and inner products below, calculate the Gram matrix.
     1. {(1 1 1 1)^T, (0 1 2 3)^T, (0 1 4 9)^T}, <x,y > = y^Tx
     2. {1, cos(x), sin(x)}, < f,g > = ∫_-π^π f(x)g(x)dx

  6. Use the Gram matrix and the inner product from (b) above to do a least-squares fit for f(x) = π − |x| on [−π,&pi] using a trigonometric polynomial T(x) = a₀ + a₁cos(x) + b₁sin(x).

Updated 4/22/07 (fjn).