M 311 Spring 2004
HOMEWORK SUMMARY REPORT
Chapter number: 6
Section number: 1
Exercise number: 4
Number of papers received: 6
Reviewing committee: Gamma
List all participating committee members:
Patrick Barrett, Mike Shelby
Author(s) of paper(s) chosen for publication: Amanda Coots
Comments:
Except for one paper, every submittal was correct. The published paper
proved/disproved every property of the inner product and was very easy to
follow. The other papers were all very well done.
GRADER'S COMMENT: I was unsatisfied (but didn't count off) with how
students proved positivity in part (a). Most got
= (x_1 + x_2)^ + 0.1 (x_2)^2 + 25 (x_3)^2
and concluded that this object is obviously positive. But the first term
can be zero even if x_1 and x_2 are nonzero. In fact, however, in that
case the (x_2)^2 term ensures positivity (if it is zero and so is the
first term, then x_1 = 0 as well as x_2), but this point is not entirely
trivial.
INSTRUCTOR'S COMMENT: This raises a more general point: In verifying the
positivity condition (" > 0 except that <0,0> = 0") in this and the
next problem, some students are stopping to prove that <0,0> = 0 (which is
ALWAYS true if the function is bilinear) and skipping over the really
important point, which is to show that is STRICTLY GREATER THAN ZERO
when x is not 0.