## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION


## DBsubject(Linear algebra)
## DBchapter(Abstract vector spaces)
## DBsection(Basis and dimension)
## Date(10/11/2013)
## Institution('NAU')
## Author('Nandor Sieben')
## RevisedBy('Sean Fitzpatrick')
## Level(2)
## MO(1)
## KEYWORDS('linear algebra','basis','subspace')

DOCUMENT();
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "MatrixReduce.pl",
  "parserMultiAnswer.pl",
  "PGcourse.pl",
  "PGML.pl", 
  "PCCmacros.pl"
);
$showPartialCorrectAnswers = 1;

sub rank {
   Context('Fraction');
   my $R = rref(apply_fraction_to_matrix_entries(shift));
   my ($row,$col) = $R->dimensions;
   my ($r,$c) = (1,1);
   while (($r <= $row) && ($c <= $col)) {
      $r++ if ($R->element($r,$c) > 0.5);
      $c ++;
   }
   $r - 1;
}

# Context("Fraction");
Context("Matrix");

$I = Matrix([[1,0],[0,1]]);
$A = Matrix([[1,0],[0,-1]]);
$B = Matrix([[0,1],[0,0]]);
$C = Matrix([[0,0],[1,0]]);

$basis = List($A, $B, $C);
$basis_checker = sub {
my ( $correct, $student, $self ) = @_;
my @errors;
my $s = @{$student};
my @c = @{$correct};
my $s0 = Matrix($s[0]); 
my $s1 = Matrix($s[1]);
my $s2 = Matrix($s[2]);
if (0 != $s0->trace) {
  push(@errors, "Your first matrix does not have trace zero.");
} 
if (0 != $s1->trace) {
  push(@errors, "Your second matrix does not have trace zero.");
} 
if (0 != $s2->trace) {
  push(@errors, "Your third matrix does not have trace zero.");
}
return (0, @errors), if @errors;

if (rank(Matrix([
    [$s0->element(1,1),$s0->element(1,2),$s0->element(2,1)],
    [$s1->element(1,1),$s1->element(1,2),$s1->element(2,1)],
    [$s2->element(1,1),$s2->element(1,2),$s2->element(2,1)]]))==3) {
    # Three matrices in basis, so score is out of 3.
    return (3, "Great work!");
}
return (2, "The matrices in your basis must be linearly independent");
};

BEGIN_PGML
Find a basis for the vector space 
[` \lbrace A\in M_{22}(\mathbb R) \mid \text{tr}(A)=0\rbrace`]
of [`2\times 2`] matrices with trace 0.

    [____________________]{$basis->cmp(list_checker => $basis_checker)}

[@KeyboardInstructions('Enter your answer as a comma-separated list of matrices. Use the syntax [|[[a,b],[c,d]]|] to enter your matrices.')@]**
END_PGML


ENDDOCUMENT();