M 311 Spring 2004 HOMEWORK SUMMARY REPORT Chapter number: 5 Section number: 2 Exercise number: 4 Number of papers received: 10 Reviewing committee: kappa List all participating committee members: Randall McClelland, Brian Young Author(s) of paper(s) chosen for publication: [James Macfarlane] Comments: All of the papers showed a good understanding of part a. Some had trouble with part b, and some just had somewhat incomplete explanations for why L was onto. Other than that, all of the papers showed a good understanding of the topic. [...]'s paper was perfect with a reference to a theorem in part b's solution, which was nice to see. INSTRUCTOR'S COMMENTS: (1) I think James's very brief argument is better than the paper chosen by the reviewers, which contains some language that could be confusing. (2) In part (a), many people left off the arbitrary constant in the specification of the (one-dimensional) kernel. If v is the only vector in a basis for the kernel, then "the kernel" is either "span (v)" or "all vectors of the form cv". (3) In (b) there are two ways of determining whether the function is onto. One (see James's paper) is just to look at the 2-row matrix and observe that its columns obviously span all of R^2. The other (which may be useful in more complicated problems) is to observe that the reduced matrix in (a) had two linearly independent rows; since the dimension of the column space is the same as that of the row space, and the latter is not changed by row reduction, this shows that the range is two-dimensional. Note, however, that if the function turns out not to be onto, then the span of the columns of the REDUCED matrix is not necessarily a basis for the range of the original matrix!