DOCUMENT();

loadMacros(
	'PGstandard.pl',   'MathObjects.pl', 'PGML.pl', 'contextFraction.pl',
	'MatrixReduce.pl', 'PGcourse.pl'
);

$showPartialCorrectAnswers = 1;

TEXT(beginproblem());

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

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
		);
}

$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\} .```]

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

[` X =`] [@ $answer->ans_array @]*

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

[` Y = `] [@ $answer2->ans_array @]*
END_PGML

ANS($A1);
ANS($A2);

ENDDOCUMENT();