Math 817: ``All about (Gram-)Schmidt'' Handout Solutions

  1. Let Pn be the vector space of all real polynomials of the form a0 + a1x + ... + anxn. Given f(x) and g(x) in Pn, define the symmetric bilinear form f*g by integrating f(x)g(x) from -1 to 1. Find an orthonormal basis for Pn for n = 1, 2 and 3.
    Solution: A basis for Pn is given by 1, x, x2, ..., xn. Now we find an orthogonal basis, which in the end we normalize to get an orthonormal basis. (The resulting polynomials, up to scalar multiples, are called Legendre polynomials. They also come up in differential equations. You can google them to find out more.) So we take u1 = 1; then 1*1 = c1 = 2. Next, u2 = x - (1/c1)(x*1)1 = x, since 1*x = 0. Also, c2 = x*x = 2/3. Next, u3 = x2 - (1/c2)(x2*x)x - (1/c1)(x2*1)1 = x2 - 0x - (1/2)(2/3)1 = x2 - (1/3); to avoid fractions, we'll use u3 = 3x2 - 1, instead. Now c3 = 8/5. And finally u4 = x3 - (1/c3)(x3*(3x2 - 1))(3x2 - 1) - (1/c2)(x3*x)x - (1/c1)(x3*1)1 = x3 - 0x2 - (3/5)x - 0 = x3 - (3/5)x. As usual, we can take 5x3 - 3x, instead of x3 - (3/5)x; then c4 = 8/7. Normalize by dividing ui by the square root of ci. Normalizing u1 through un+1 gives an orthonormal basis of Pn. (Thus we don't need to start from scratch to do P3, after having done P2. We just need to do the next vector.)