View of /branches/UGA/4.4.3a.pg

    1 ## DESCRIPTION
    2 ##   Orthogonal complement
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Linear equations')
    6 ##
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('')
   10 ## DBsection('')
   11 ## Date('10/05/2009')
   12 ## Author('Ted Shifrin')
   13 ## Institution('UGA')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT(); # This should be the first executable line in the problem.
   21 
   22 loadMacros("PG.pl",
   23            "PGbasicmacros.pl",
   24            "PGchoicemacros.pl",
   25            "PGanswermacros.pl",
   26            "PGauxiliaryFunctions.pl",
   27            "PGmatrixmacros.pl",
   28            "PGmorematrixmacros.pl",
   29            "compoundProblem.pl",
   30           );
   31 
   32 TEXT(beginproblem());
   33 $showPartialCorrectAnswers = 1;
   34 
   35 $a = non_zero_random(-2, 2, 1);
   36 $b = non_zero_random(-2, 2, 1);
   37 $c = non_zero_random(-2, 2, 1);
   38 $d = non_zero_random(-3, 3, 1);
   39 $e = non_zero_random(-3, 3, 1);
   40 $f = non_zero_random(-3, 3, 1);
   41 
   42 
   43 $i = non_zero_random(-2, 2, 1);
   44 $j = non_zero_random(-1,1,2);
   45 
   46 $k = $i;
   47 $l = $j;
   48 $m = $i*$a+$j*$d;
   49 $n = $i*$b+$j*$e;
   50 $o = $i*$c+$j*$f;
   51 
   52 if($a==1){$aa="";} elsif ($a==-1){$aa="-";} else{$aa=$a;}
   53 if($b==1){$bb="";} elsif ($b==-1){$bb="-";} else{$bb=$b;}
   54 if($c==1){$cc="";} elsif ($c==-1){$cc="-";} else{$cc=$c;}
   55 if($k==1){$kk="";} elsif ($k==-1){$kk="-";} else{$kk=$k;}
   56 if($l==1){$ll="";} elsif ($l==-1){$ll="-";} else{$ll=$l;}
   57 if($m==1){$mm="";} elsif ($m==-1){$mm="-";} else{$mm=$m;}
   58 if($n==1){$nn="";} elsif ($n==-1){$nn="-";} else{$nn=$n;}
   59 if($o==1){$oo="";} elsif ($o==-1){$oo="-";} else{$oo=$o;}
   60 
   61 $isProfessor = ($studentLogin eq 'shifrin' || $studentLogin eq 'test');
   62 
   63 $cp = new compoundProblem(
   64   parts=>2,
   65   weights=>[.2,.8],
   66   parserValues=>1,
   67   allowReset => $isProfessor,
   68   nextVisible => 'Always',
   69   nextStyle => 'Button',
   70 );
   71 
   72 $part = $cp->part;
   73 
   74 if($part==1){
   75 
   76 BEGIN_TEXT
   77 
   78 Suppose
   79 \[ V = \text{Span}\left(\left[\begin{array}{r} 1\cr 0\cr $a\cr $b\cr $c
   80 \end{array}\right], \left[\begin{array}{r} $k\cr $l\cr $m\cr $n\cr $o\end{array}\right]\right)\subset \mathbb R^5\ . \]
   81 
   82 $PAR
   83 dim \(V\) = \{ans_rule(3)\}
   84 
   85 $PAR
   86 Now, let
   87 
   88 \[ W = \left\{"\{"\} \left[\begin{array}{c} x_1\cr x_2\cr x_3\cr x_4\cr x_5\end{array}\right]\in\mathbb R^5: \begin{array}{r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r}
   89   x_1   &           & + $aa x_3 & + $bb x_4 & + $cc x_5 & = & 0 \cr
   90 $kk x_1 & + $ll x_2 & + $mm x_3 & + $nn x_4 & + $oo x_5 & = & 0
   91 \end{array} \right\} \ . \]
   92 
   93 $PAR
   94 dim \(W\) = \{ans_rule(3)\}
   95 
   96 END_TEXT
   97 ANS(num_cmp(2));
   98 ANS(num_cmp(3));
   99 }
  100 
  101 if($part==2){
  102 BEGIN_TEXT
  103 
  104 Recall that
  105 
  106 \[ V = \text{Span}\left(\left[\begin{array}{r} 1\cr 0\cr $a\cr $b\cr $c
  107 \end{array}\right], \left[\begin{array}{r} $k\cr $l\cr $m\cr $n\cr $o\end{array}\right]\right)\subset \mathbb R^5\ . \]
  108 
  109 $PAR
  110 Give a basis for \( V^\perp \).
  111 
  112 \{mbox( ans_array(5,1,8), ',', ans_array_extension(5,1,8),',', ans_array_extension(5,1,8) ) \}
  113 
  114 $PAR
  115 
  116 Recall that
  117 
  118 \[ W = \left\{"\{"\} \left[\begin{array}{c} x_1\cr x_2\cr x_3\cr x_4\cr x_5\end{array}\right]\in\mathbb R^5: \begin{array}{r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r${NO_SPACE}r}
  119   x_1   &           & + $aa x_3 & + $bb x_4 & + $cc x_5 & = & 0 \cr
  120 $kk x_1 & + $ll x_2 & + $mm x_3 & + $nn x_4 & + $oo x_5 & = & 0
  121 \end{array} \right\} \ . \]
  122 
  123 $PAR
  124 Give a basis for \( W^\perp \).
  125 
  126 \{mbox( ans_array(5,1,8),',', ans_array_extension(5,1,8) ) \}
  127 
  128 END_TEXT
  129 
  130 ANS(basis_cmp([[-$a,-$d,1,0,0],[-$b,-$e,0,1,0],[-$c,-$f,0,0,1]]));
  131 ANS(basis_cmp([[1,0,$a,$b,$c],[0,1,$d,$e,$f]]));
  132 }
  133 
  134 
  135 ENDDOCUMENT();  # This should be the last executable line in the problem.
  136 
  137