Math 603-601 - Fall 2002

Assignment 5

1. Derive the formulas on pg. 46 for i'₁, i'₂, and i'₃ in terms of i₁, i₂, and i₃ by multiplying out the three rotation matrices making up the change of basis matrix A we derived in class. The entries of the transpose of A are the coefficients you will need.

2. If A and B are orthogonal matrices, then show that both AB and BA are orthogonal matrices as well.

3. Use the Gram-Schmidt process to turn {1,x,x²} into an orthonormal set relative the inner product < f, g > = ₀S ¹ f(x)g(x)dx.

4. Find the QR factorization for the matrix A given below.

   1  -1   2
   1   2  -1
   0   1   1
   2   1   1
   

5. Find and sketch the discrete least squares fit to the data in the table below.

   Log of Concentration

   t	0	1	2	3	4
   ln(C)	-0.1	-0.4	-0.8	-1.1	-1.5

6. Use the orthonormal basis you found in problem 3 and class notes on finding the minimizer to obtain the polynomial in P₂ that gives the best continuous least square fit for the function f(x) = e^2x on the interval [0,1]. Sketch both the function and the polynomial that fits it.