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!