## DBsubject(Linear algebra)
## DBchapter(Abstract vector spaces)
## DBsection(Span)
## Date(10/5/2013)
## Institution(NAU)
## Author(Nandor Sieben)
## RevisedBy('Sean Fitzpatrick')
## Level(2)
## MO(1)
## KEYWORDS('linear algebra','span')

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

Context("Fraction");
Context() -> parens -> set ("[" => {formMatrix => 1});

## include rank algorithm from rank.pl file

sub rank {
	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;
}

{
$a11= random(-5,5,1);
$a12= random(-5,5,1);
$a22= random(-5,5,1);
$b11= random(-5,5,1);
$b12= random(-5,5,1);
$b22= random(-5,5,1);
Context() -> parens -> set ("[" => {formMatrix => 1});
$A = Matrix([[$a11,$a12],[$a12,$a22]]);
$B = Matrix([[$b11,$b12],[$b12,$b22]]);
redo if(rank(Matrix([[$a11,$a12,$a22],[$b11,$b12,$b22],[0,0,0]]))<2);
}
$answer = $A+$B;
$answer = $A-$B if (rank($answer) == 0);

{
$c1 = Compute(random(-2,2,1));
$c2 = Compute(random(-2,2,1));
$c3 = Compute(1);
Context() -> parens -> set ("[" => {formMatrix => 1});
$answer2=Matrix([[$c1,$c3],[$c3,$c2]]);
redo if (rank(Matrix([$a11,$a12,$a22],[$b11,$b12,$b22],
      [$answer2->element(1,1),$answer2->element(1,2),$answer2->element(2,2)])) == 2);
}

Context() -> normalStrings;

$A1 = $answer->cmp(
  checker => sub {
  my ($correct, $student, $ansHash) = @_;
  if ( ! $student->is_symmetric )
    { return 0; } 
  if ( rank(Matrix($student)) == 0 )
    { return 0; } 
  if (rank(Matrix([$a11,$a12,$a22],[$b11,$b12,$b22],
      [$student->element(1,1),$student->element(1,2),$student->element(2,2)])) == 2) {
    return 1;
  }
  return 0;
  }
);

$A2 = $answer2->cmp(
  checker => sub {
  my ($correct, $student, $ansHash) = @_;
  if ( ! $student->is_symmetric )
    { return 0; } 
  if (rank(Matrix([$a11,$a12,$a22],[$b11,$b12,$b22],
      [$student->element(1,1),$student->element(1,2),$student->element(2,2)]))==3) {
    return 1;
  }
  return 0;
  }
);


BEGIN_PGML
Let [`V`] be the vector space of symmetric [`2\times 2`] matrices and
[`W`] be the subspace 
[```W=\operatorname{span} \left\{ [$A],[$B] \right\} .```]

1. Find a nonzero element [`X`] in [` W `].

    [__________]{$A1}

2. Find an element [`Y`] in [` V `] that is not in  [` W `].

    [__________]{$A2}

    [@KeyboardInstructions('Enter your matrices using the syntax [|[[a,b],[c,d]]|].')@]**

END_PGML




ENDDOCUMENT();