M 311 Spring 2004
HOMEWORK SUMMARY REPORT
Chapter number: 6
Section number: 1
Exercise number: 3
Number of papers received: 4
Reviewing committee: Theta
List all participating committee members:
Xujia Zhang, Jason Madsen
Author(s) of paper(s) chosen for publication: Kathryn Turner
Comments:
We received four papers for this problem. Two of the solutions have some
minor mathmatically mistakes (left out the square root signs in the second
step for both of them). The other two are correct (one of them needs more
mathmatically derviations and illustrations). We chose to publish Kathryn
Turner's paper because hers is very well written and easy to follow.
GRADER'S COMMENT: Several people seemed to think that bilinearity implies
= + .
Instead, using bilinearity twice, you have
= +
= + 2 + .
INSTRUCTOR'S COMMENT: Most papers were organized by writing down the
inequality to be proved and then manipulating it into something that is
obviously true. That is probably the easiest way to FIND a proof;
however, PRESENTING the proof that way can be confusing, for two reasons.
One, it may not be clear to the reader (unless you write "?" above the
inequality signs) which statements are being investigated rather than
asserted. Second, both you and the reader may get mixed up about the
direction of the logic. In the present case, the inequality to be proved
can be shown to be equivalent to the Cauchy-Schwarz inequality without
absolute value signs on the left side. This does not imply the inequality
with the absolute value, but it is IMPLIED BY it, which is what is needed
to prove the theorem!