Math 603-601 - Fall 2002

Homework

Assignment 4

  1. Consider the space P2 of degree 2 polynomials. Let B = {2, (3/2)½ x, (5/2)½ (½)(3x2 - 1)}.
    1. Verify that B is an orthonormal set, given that the inner product is

    2. Let p(x)=x2 + x -2 and q(x) = 2x2 +3x -4. Using the formula for the coordinates derived in class, find the coordinate vectors [p]B and [q]B.
    3. Use your answer to part (a) along with the formula < p, q > = [q]BT [p]B to find the inner product < p, q >.
    4. Compare the value you got for < p, q > in part (b) with the one that you obtain by doing the integral in part (a) to get the inner product.

  2. Consider the inner product defined by

    Recall that we showed that the set

    is orthonormal relative to this inner product. Using the same formula for coordinates that you used in the previous problem, find the "coordinates" for f(x) = ex. (You are really finding a Fourier series here.)

  3. Show that if A is a 2×2 orthogonal matrix, then either A =
    cos(t)  -sin(t)
    sin(t)  cos(t)
    

    or A =
    cos(t)  sin(t)
    sin(t)  -cos(t)
    
  4. Consider the vectors A = 2i + j - k, B = i + j + k, C = i - j - k.
    1. Show that A, B, C form a basis in 3D space.
    2. Find the area of each of the faces of the parallelepiped with edges A, B, C.
    3. Find the volume of the parallelepiped with edges A, B, C.
    4. There is a single constant k such that the three vectors a, b, c defined by
      a = kB×C, b = kC×A, c = kA×B,
      satsify these equations:
      a·A = 1   a·B = 0   a·C = 0
      b·A = 0   b·B = 1   b·C = 0
      c·A = 0   c·B = 0   c·C = 1.
      Find k.
    5. The vectors a, b, c are called a reciprocal basis to A, B, C. In terms of the parallelepiped with edges A, B, C, give a geometrical interpretation to both the direction and length of each of these vectors.

  5. Consider a 3D rotation in which k' = (i + 2j + 2k)/3. Find the angle of precession and the angle of nutation.
Due Thursday, 3 October.