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.