[LON-CAPA-users] coding a customresponse problem using &cas()
Justin Gray
jgray at math.sfu.ca
Tue May 26 20:32:21 EDT 2015
Hi Peter,
Thank you very much for this. I tested it out and it works beautifully.
I agree that, more generally, a diagonal idempotent matrix could be
regarded as a trivial example. The purpose of the exercise that I have in
mind is to draw to students' attention that operations on matrices are much
different than operations with real numbers. For real numbers, the equation
x^2 = x is satisfied only by x = 0 and x = 1, but for matrices the equation
A^2 = A has solutions other than the additive and multiplicative identities.
Thanks,
Justin
Justin Gray | Senior Lecturer
Department of Mathematics | Simon Fraser University
8888 University Drive, Burnaby | V5A 1S6 | Canada
Tel: +1 778.782.4237
On Tue, May 26, 2015 at 8:46 AM, Peter Dencker <dencker at math.uni-luebeck.de>
wrote:
>
> 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
>
> _______________________________________________
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> LON-CAPA-users at mail.lon-capa.org
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>
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