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.
If A and B are orthogonal matrices,
then show that both AB and BA are orthogonal matrices as well.
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.
Find the QR factorization for the matrix A given below.
1 -1 2
1 2 -1
0 1 1
2 1 1
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
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) = e2x on
the interval [0,1]. Sketch both the function and the polynomial that
fits it.