## DESCRIPTION
##   Orthogonal complement
## ENDDESCRIPTION

## KEYWORDS('Linear equations')
## 

## DBsubject('Calculus')
## DBchapter('')
## DBsection('')
## Date('10/05/2009')
## Author('Ted Shifrin')
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')
           
DOCUMENT();	# This should be the first executable line in the problem.

loadMacros("PG.pl",      
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",      
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "PGmatrixmacros.pl",
           "PGmorematrixmacros.pl",
           "compoundProblem.pl",
          );      
     
TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = non_zero_random(-2, 2, 1);
$b = non_zero_random(-2, 2, 1);
$c = non_zero_random(-2, 2, 1);
$d = non_zero_random(-3, 3, 1);
$e = non_zero_random(-3, 3, 1);
$f = non_zero_random(-3, 3, 1);


$i = non_zero_random(-2, 2, 1);
$j = non_zero_random(-1,1,2);

$k = $i;
$l = $j;
$m = $i*$a+$j*$d;
$n = $i*$b+$j*$e;
$o = $i*$c+$j*$f;

if($a==1){$aa="";} elsif ($a==-1){$aa="-";} else{$aa=$a;}
if($b==1){$bb="";} elsif ($b==-1){$bb="-";} else{$bb=$b;}
if($c==1){$cc="";} elsif ($c==-1){$cc="-";} else{$cc=$c;}
if($k==1){$kk="";} elsif ($k==-1){$kk="-";} else{$kk=$k;}
if($l==1){$ll="";} elsif ($l==-1){$ll="-";} else{$ll=$l;}
if($m==1){$mm="";} elsif ($m==-1){$mm="-";} else{$mm=$m;}
if($n==1){$nn="";} elsif ($n==-1){$nn="-";} else{$nn=$n;}
if($o==1){$oo="";} elsif ($o==-1){$oo="-";} else{$oo=$o;}

$isProfessor = ($studentLogin eq 'shifrin' || $studentLogin eq 'test');

$cp = new compoundProblem(
  parts=>2,
  weights=>[.2,.8],
  parserValues=>1,
  allowReset => $isProfessor,
  nextVisible => 'Always',
  nextStyle => 'Button',
);

$part = $cp->part;

if($part==1){

BEGIN_TEXT

Suppose
\[ V = \text{Span}\left(\left[\begin{array}{r} 1\cr 0\cr $a\cr $b\cr $c
\end{array}\right], \left[\begin{array}{r} $k\cr $l\cr $m\cr $n\cr $o\end{array}\right]\right)\subset \mathbb R^5\ . \]

$PAR
dim \(V\) = \{ans_rule(3)\}

$PAR
Now, let

\[ W = \left\{"\{"\} \left[\begin{array}{c} x_1\cr x_2\cr x_3\cr x_4\cr x_5\end{array}\right]\in\mathbb R^5: \begin{array}{r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r}
  x_1   &           & + $aa x_3 & + $bb x_4 & + $cc x_5 & = & 0 \cr
$kk x_1 & + $ll x_2 & + $mm x_3 & + $nn x_4 & + $oo x_5 & = & 0
\end{array} \right\} \ . \] 

$PAR
dim \(W\) = \{ans_rule(3)\}

END_TEXT
ANS(num_cmp(2));
ANS(num_cmp(3));
}

if($part==2){
BEGIN_TEXT

Recall that

\[ V = \text{Span}\left(\left[\begin{array}{r} 1\cr 0\cr $a\cr $b\cr $c
\end{array}\right], \left[\begin{array}{r} $k\cr $l\cr $m\cr $n\cr $o\end{array}\right]\right)\subset \mathbb R^5\ . \]

$PAR
Give a basis for \( V^\perp \).

\{mbox( ans_array(5,1,8), ',', ans_array_extension(5,1,8),',', ans_array_extension(5,1,8) ) \}

$PAR

Recall that

\[ W = \left\{"\{"\} \left[\begin{array}{c} x_1\cr x_2\cr x_3\cr x_4\cr x_5\end{array}\right]\in\mathbb R^5: \begin{array}{r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r}
  x_1   &           & + $aa x_3 & + $bb x_4 & + $cc x_5 & = & 0 \cr
$kk x_1 & + $ll x_2 & + $mm x_3 & + $nn x_4 & + $oo x_5 & = & 0
\end{array} \right\} \ . \] 

$PAR
Give a basis for \( W^\perp \).

\{mbox( ans_array(5,1,8),',', ans_array_extension(5,1,8) ) \}

END_TEXT

ANS(basis_cmp([[-$a,-$d,1,0,0],[-$b,-$e,0,1,0],[-$c,-$f,0,0,1]]));
ANS(basis_cmp([[1,0,$a,$b,$c],[0,1,$d,$e,$f]]));
}


ENDDOCUMENT();	# This should be the last executable line in the problem.