[LON-CAPA-users] coding a customresponse problem using &cas()

[Peter Dencker]

Hi Justin,

in my example a diagonal idempotent matrix will be regarded as too trivial.

- Peter


<problem>

<script type="loncapa/perl">

     sub is_true_but_diagonal {
         my @given = @_;
         # is the matrix diagonal?
         for my $i ( 0 .. $#given ) {
             for my $j ( 0 .. $#{ $given[$i] } ) {
                 if ( $i != $j && $given[$i][$j] != 0 ) {
                     return 0;
                 }
             }
         }
         # is the diagonal matrix idempotent?
         for my $i ( 0 .. $#given ) {
             if ( $given[$i][$i] != 0 && $given[$i][$i] != 1 ) {
                 return 0;
             }
         }
         return 1;
     }

     $hint = q{};

</script>

Give a nontrivial example of an idempotent matrix (whose entries are all 
integers). <br />


<customresponse id="0r0">
   <answer type="loncapa/perl">

        for ($submission) { s{\s}{}gxms; s{\Amatrix}{}gxms; }

         return 'BAD_FORMULA'
             if $submission !~ /\A[(]\[(.+)\][)]\z/xms;
         my @given
           = map { [ split /,/xms, $_ ]; } split /\],\[/xms, $1;
         for (map { @{$_}; } @given) {
             return 'BAD_FORMULA' if !/\A[-+]?\d+\z/xms;
         }

         my $n = @{ $given[0] };
         for my $row (@given) { return 'BAD_FORMULA' if @{$row} != $n }
         my $m = @given;
         return 'INCORRECT' if $m != $n;

         return 'BAD_FORMULA' if is_true_but_diagonal(@given);

         my $matrix      = "matrix$submission";
         my $maxima_in
           = "is(rank(rat($matrix.$matrix-$matrix))=0)"
           . q{;};
         my $maxima_out = cas( 'maxima', $maxima_in );
         return 'EXACT_ANS'
             if $maxima_out eq 'true';
         return 'INCORRECT'
             if $maxima_out eq 'false';
         return 'BAD_FORMULA';

     </answer>
     <customhint id="0r0h0">
       <answer type="loncapa/perl">

           for ($submission) { s{\s}{}gxms; s{\Amatrix}{}gxms; }
           if ( $submission !~ /\A[(]\[(.+)\][)]\z/xms ) {
               $hint = '<b>Give a matrix in the required form.</b>';
               return;
           }
           my @given
             = map { [ split /,/xms, $_ ]; } split /\],\[/xms, $1;
           for ( map { @{$_}; } @given ) {
               if ( !/\A[-+]?\d+\z/xms ) {
                   $hint
                     = '<b>Use integer entries for this submission.</b>';
                   return;
               }
           }
           my $n = @{ $given[0] };
           for my $row (@given) {
               if ( @{$row} != $n ) {
                   $hint
                     = '<b>All rows must be the same length.<b>';
                   return;
               }
           }
           my $m = @given;
           if ( $m != $n ) {
               $hint
                 = '<b>An idempotent matrix must necessarily'
                 . 'be a square matrix!</b>';
               return;
           }
           if ( is_true_but_diagonal(@given) ) {
               $hint
                 = '<b>Correct, well, but choose'
                 . 'a less trivial example.</b>';
               return;
           }

       </answer>
     </customhint>
</customresponse>

<textline /> <br />
$hint <br />

</problem>




Am 05/26/2015 um 03:49 AM schrieb Justin Gray:
> I would appreciate it if someone could assist me with coding the following
> problem.
>
> Give an example of an idempotent matrix.
>
> Ideally, I would like students to input their answer using the format
> (row_1,...,row_m) using a matrix of any size, so that ([1,1],[0,0]) and
> ([2,-2,-4],[-1,3,4],[1,-2,-3]) would both be acceptable answers.
> (A matrix *A* is *idempotent* if *A*^2=*A*.)
>
> If I understand correctly, using a mathresponse problem is problematic in
> this case because there is some preprocessing of the students submission
> that makes it difficult to use in the answer algorithm, but one way around
> this is to use customresponse combined with the &cas() function.
>
> In order that Maxima understands the students submission, the expression
> 'matrix' needs to be appended to the front. Also, matrix exponentiation is
> denoted by A^^n in Maxima. If it is easier to test a numerical condition,
> one could verify that rank(A^^2 - A) = 0.
>
> Thanks,
> Justin
>
> P.S. Ideally, I would like to stipulate that students provide a nontrivial
> example (excluding the zero matrix and the identity matrix) but that is the
> topic of another discussion.
>
>
>
> Justin Gray | Senior Lecturer
> Department of Mathematics | Simon Fraser University
> 8888 University Drive, Burnaby | V5A 1S6 | Canada
> Tel: +1 778.782.4237
>
>
>
> _______________________________________________
> LON-CAPA-users mailing list
> LON-CAPA-users at mail.lon-capa.org
> http://mail.lon-capa.org/mailman/listinfo/lon-capa-users
>


-- 
Dr. Peter Dencker
     wissenschaftl. Mitarbeiter

UNIVERSITÄT ZU LÜBECK
     INSTITUT FÜR MATHEMATIK

     Ratzeburger Allee 160
     23562 Lübeck

     Tel +49 451 500 4254
     Fax +49 451 500 3373
     dencker at math.uni-luebeck.de

     www.math.uni-luebeck.de