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

DOCUMENT();
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "AnswerFormatHelp.pl",
  "MatrixReduce.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;
}

$l=non_zero_random(-9,9,1);
$p1=Formula("2*$l-1+x^2")->reduce;
$p2=Formula("x")->reduce;

$basis = List($p1, $p2);
$listChecker = sub {
my ( $correct, $student, $self ) = @_;
my @errors;
my ($s1,$s2) = @{$student};
my @c = @{$correct};

my $s1D=$s1->D('x');
my $s2D=$s2->D('x');
my $s1DD=$s1D->D('x');
my $s2DD=$s2D->D('x');

if ($s1D->eval(x=>$l) != $s1->eval(x=>1)) {
  push(@errors, "Your first polynomial does not have the right derivative.");
} 

if ($s2D->eval(x=>$l) != $s2->eval(x=>1)) {
  push(@errors, "Your second polynomial does not have the right derivative.");
}

my $s1DDD=$s1DD->D('x');
my $s2DDD=$s2DD->D('x');
if ($s1DDD != Formula("0")) {
  push(@errors, "Your first polynomial is not in `P_2(\mathbb{R})`.");
}

if ($s2DDD != Formula("0")) {
  push(@errors, "Your second polynomial is not in `P_2(\mathbb{R})`.");
}
return (0, @errors) if @errors;

my $a1=$s1->eval(x=>0);
my $b1=$s1D->eval(x=>0);
my $c1=$s1DD->eval(x=>0)/2;
my $a2=$s2->eval(x=>0);
my $b2=$s2D->eval(x=>0);
my $c2=$s2DD->eval(x=>0)/2;
if (rank(Matrix([$a1,$b1,$c1],[$a2,$b2,$c2])) ==2) {
     # correct answers list consists of 2 items, so score is out of 2 total.
    return (2, "Good job!");
}
return (1, "A basis must consist of linearly independent elements.");
};


BEGIN_PGML

Find a basis [` \lbrace p(x), q(x) \rbrace `] for the vector space 
[`\lbrace f(x)\in P_2(\mathbb{R}) \mid f'([$l])=f(1) \rbrace`] 
where [`P_2(\mathbb{R})`] is the vector space of polynomials in [`x`] with degree less than or equal to 2.

    [__________]{$basis->cmp(list_checker => $listChecker)}

[@KeyboardInstructions('Enter your answer as a comma-separated list of polynomials.')@]**

END_PGML

ENDDOCUMENT();