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Please show all your work! Total points: (longer than an actual exam)
  1. Consider the space P2 with inner product defined as <p(x),q(x)> = p(0)q(0). Is this an inner product space?
    No, does not satisfy positivity: p(x) = x has zero length according to this inner product, but p(x) ≠ 0.
  2. Consider the space F[0,∞) with inner product defined as <f,g> = ∫0∞ f(x)g(x)dx.
    1. Prove that this is an inner product space.
      It is okay to assume that F[0,1] is a vector space, and just prove the four axioms on the inner product.
    2. Let fn(x) = e-nx. Find the norm of fn for any n>0.
      ∫ e-2nxdx = (-1/2n) e-nx.
      Eval at [0,∞): ||fn||2 = 1/2n
      So ||fn|| = 1/√(2n).
    3. What about fn for n≤0? Describe the graph of the functions, and find their norms.
      At n=0, f0 = 1 (constant function), which has infinite norm in this space. For n<0, fn increases without bound (exponential growth), so the norm is also infinite.
    4. Apply the Gram-Schmidt process to obtain an orthonormal set from the set {f1, f2}.
      Let v1 = f1; according to a previous part, its norm is 1/√(2n) = 1/√2, so the normalized version is q1 = √2 e-x. Is f2 orthogonal to q1? Their inner product is √2 ∫ e-3xdx = √2 /3 ≠ 0, so no.
      The projection of f2 onto q1 is (2/3)e-x, so the orthogonal complement is v2 = f2 - proj(f2 onto q1) = e-2x - (2/3)e-x.
      To normalize this, we need to find its norm:
      ||v2||2 = ∫ ( e-2x - (2/3)e-x )2 dx = ∫ ( e-4x - (4/3)e-3x + (4/9)e-2x ) dx = (1/4) - (4/9) + (2/9) = 1/36.
      So ||v2|| = 1/6, so q2 = 6v2 = 6 e-2x - 4 e-x.
      The orthonormal set is {q1, q2} = { √2 e-x, 6 e-2x - 4 e-x }.
    5. Is this set an orthonormal basis?
      Not for all of F[0,∞), which is an infinite-dimensional space.
  3. Find a QR-decomposition for the matrix, if possible:
    101
    -111
    101
    -111
    .
  4. Consider the set of points (0,1), (1,3), (2,2), (3,3).
    1. Find the polynomial of lowest degree which interpolates these points.
    2. Find the line which best fits these points using least-squares approximation.
    3. What is the error in this approximation? (i.e., ||Ax-b||).
    4. Find the quadratic (parabola) which best fits these points using least-squares approximation.
    5. What is the error in this approximation? (i.e., ||Ax-b||).
  5. Consider the matrix A:
    1 0 0
    0 -1 2
    0 2 2
    1. Does there exist an orthogonal matrix P for which P-1AP is a diagonal matrix?
      Yes, A is orthogonally diagonalizable, because it is symmetric.
    2. Find all eigenvalues and eigenvectors of A.
      @λ=1: x = (1,0,0)t. @λ=3: x = (0,1/2,1)t. @λ=-2: x = (0,-1/2,1)t.
    3. Find an orthogonal matrix P which diagonalizes A.
      Just normalize the eigenvectors:
      1 0 0
      0 1/√5-1/√5
      0 2/√5 2/√5