M 311 Spring 2004
HOMEWORK SUMMARY REPORT
Chapter number: 6
Section number: 2
Exercise number: 4
Number of papers received: 2
Reviewing committee: alpha
List all participating committee members:
Tristan Brown, James Macfarlane
Author(s) of paper(s) chosen for publication: Kathryn Turner
Comments:
Both students seem to understand the basic idea for the Gram-Schmit
process. The only problems seemed to be small algebraic mistakes and
inconsistent labeling.
GRADER'S COMMENT: Neither paper normalized the vectors.
INSTRUCTOR'S COMMENT: The problem called for "orthogonal" polynomials,
not "orthonormal" ones, so technically it is correct to provide
polynomials that are not of unit length; in fact, a common convention is
to choose "monic" orthogonal polynomials (i.e., choose the coefficient of
the highest-degree term to be equal to 1). HOWEVER, the results then will
not be correct unless you divide each term in the projection formula by
the square of the norm of the basis vector! For example, if v_1 is not
of unit length, then the parallel part of v_2 (the projection of v_2
onto v_1) is
v_1 / ||v_1||^2 ,
and the new basis vector orthogonal to v_1 is v_2 minus that.
The details of Kathryn's calculation on p. 2 are not right (for instance,
u_0 = (e^2 - 1)^{-1/2}, not (e^2 - 1)^{+1/2}). However, her exposition
and organization of the overall process are very good. The algebra of a
correct calculation is in fact very complicated, and I'll change this
problem in the next edition of the book.