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_2> 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.