## 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",
  "parserMultiAnswer.pl",
  "AnswerFormatHelp.pl",
  "MatrixReduce.pl",
  "rank.pl",
  "PGcourse.pl"
);
$showPartialCorrectAnswers = 1;
TEXT(beginproblem()); 

Context("Numeric");

$l=non_zero_random(-9,9,1);

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

$multians = MultiAnswer($p1, $p2)->with(
singleResult => 1,
checker => sub {
my ( $correct, $student, $self ) = @_;
my ($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)) {
  $self->setMessage(1,"Not in the subspace.");
  return 0 
} 
if ($s2D->eval(x=>$l) != $s2->eval(x=>1)) {
  $self->setMessage(2,"Not in the subspace.");
  return 0 
} 
my $s1DDD=$s1DD->D('x');
my $s2DDD=$s2DD->D('x');
if ($s1DDD != Formula("0")) {
  $self->setMessage(1,"Not in the subspace.");
  return 0 
} 
if ($s2DDD != Formula("0")) {
  $self->setMessage(2,"Not in the subspace.");
  return 0 
} 
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) {
    return 1;
}
$self->setMessage(1,"Not independent.");
$self->setMessage(2,"Not independent.");
return 0;
}
);

Context()->texStrings;
BEGIN_TEXT
Find a basis \( \lbrace p(x), q(x) \rbrace \) for the vector space 
\(\lbrace f(x)\in{\mathbb P}_3[x] \mid f'($l)=f(1) \rbrace\) 
where \({\mathbb P}_3[x]\) is the vector space of polynomials in \(x\) with degree less than 3.
$BR
\( p(x) = \)
\{ $multians->ans_rule(15) \}, 
\( q(x) = \)
\{ $multians->ans_rule(15) \}


END_TEXT
Context()->normalStrings;

ANS( $multians->cmp() );

;

ENDDOCUMENT();