View of /branches/UGA/4.4.2.pg

    1 ## DESCRIPTION
    2 ##   Fundamental Subspaces of a Matrix
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Nullspace', 'Column space', 'Row space', 'Left nullspace')
    6 ##
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('')
   10 ## DBsection('')
   11 ## Date('10/04/2009')
   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("PGstandard.pl",
   23            "PGbasicmacros.pl",
   24            "PGchoicemacros.pl",
   25            "PGanswermacros.pl",
   26            "PGauxiliaryFunctions.pl",
   27            "PGmatrixmacros.pl",
   28            "PGmorematrixmacros.pl",
   29            "compoundProblem.pl",
   30            "PGnumericalmacros.pl",
   31           );
   32 
   33 
   34 TEXT(beginproblem());
   35 $showPartialCorrectAnswers = 1;
   36 
   37 $a = non_zero_random(-3, 3, 1);
   38 $b = non_zero_random(-3, 3, 1);
   39 $c = non_zero_random(-3, 3, 1);
   40 $d = non_zero_random(-3, 3, 1);
   41 $d1 = non_zero_random(-3, 3, 1);
   42 $e = non_zero_random(-3, 3, 1);
   43 $f = non_zero_random(-3, 3, 1);
   44 $i = non_zero_random(-1, 1, 2);
   45 $g = $e*$f+$i;
   46 $k = non_zero_random(-3, 3, 1);
   47 $l = non_zero_random(-3, 3, 1);
   48 $m = non_zero_random(-3, 3, 1);
   49 $n = non_zero_random(-3, 3, 1);
   50 
   51 
   52 $m11=1; $m12=$a; $m13=$e;  $m14=$b+$e*$d; $m15=$c+$e*$d1;
   53 $m21=$f; $m22=$f*$a; $m23=$g; $m24=$f*$b+$g*$d; $m25=$f*$c+$g*$d1;
   54 $m31=$k; $m32=$k*$a; $m33=$l; $m34=$k*$b+$l*$d; $m35=$k*$c+$l*$d1;
   55 $m41=$m; $m42=$m*$a; $m43=$n; $m44=$m*$b+$n*$d; $m45=$m*$c+$n*$d1;
   56 
   57 $LN11=-$k+$i*$f*($l-$e*$k);
   58 $LN12=-$i*($l-$e*$k);
   59 $LN13=1;
   60 $LN14=0;
   61 
   62 $LN21=-$m+$i*$f*($n-$e*$m);
   63 $LN22=-$i*($n-$e*$m);
   64 $LN23=0;
   65 $LN24=1;
   66 
   67 $isProfessor = ($studentLogin eq 'shifrin' || $studentLogin eq 'test');
   68 
   69 $cp = new compoundProblem(
   70   parts=>2,
   71   weights=>[.4,.6],
   72   parserValues=>1,
   73   allowReset => $isProfessor,
   74   nextVisible => 'Always',
   75   nextStyle => 'Button',
   76 );
   77 
   78 $part = $cp->part;
   79 
   80 if($part==1){
   81 BEGIN_TEXT
   82 
   83 Consider the matrix
   84 
   85 \[ A = \left[\begin{array}{r r r r r}
   86 $m11 & $m12 & $m13 & $m14 & $m15\cr
   87 $m21 & $m22 & $m23 & $m24 & $m25 \cr
   88 $m31 & $m32 & $m33 & $m34 & $m35 \cr
   89 $m41 & $m42 & $m43 & $m44 & $m45
   90 \end{array} \right]\quad .
   91 \]
   92 
   93 $PAR
   94 We start with some basic facts about the four fundamental subspaces
   95 associated to \(A\):
   96 $PAR
   97 \(\mathbf R(A) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
   98 $BR
   99 \(\mathbf C(A) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
  100 $BR
  101 \(\mathbf N(A) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
  102 $BR
  103 \(\mathbf N(A^T) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
  104 
  105 $PAR
  106 Give the reduced echelon form of \(A\).
  107 $BR
  108 \{ mbox( answer_matrix(4,5,5) )\}
  109 
  110 
  111 $PAR
  112 Give the dimension of each of the fundamental subspaces:
  113 $BR
  114 dim \(\mathbf R(A) = \) \{ ans_rule(5) \}
  115 $BR
  116 dim \(\mathbf C(A) = \) \{ ans_rule(5) \}
  117 $BR
  118 dim \(\mathbf N(A) = \) \{ ans_rule(5) \}
  119 $BR
  120 dim \(\mathbf N(A^T) = \) \{ ans_rule(5) \}
  121 $PAR
  122 
  123 When you have answered all these questions correctly, click on the "Go on to next part" button.
  124 
  125 END_TEXT
  126 
  127 ANS(num_cmp(5)); ANS(num_cmp(4)); ANS(num_cmp(5)); ANS(num_cmp(4));
  128 
  129 ANS(num_cmp(1)); ANS(num_cmp($a)); ANS(num_cmp(0)); ANS(num_cmp($b)); ANS(num_cmp($c));
  130 ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(1)); ANS(num_cmp($d));
  131 ANS(num_cmp($d1));
  132 ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0));
  133 ANS(num_cmp(0));
  134 ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0));
  135 ANS(num_cmp(0));
  136 
  137 ANS(num_cmp(2)); ANS(num_cmp(2)); ANS(num_cmp(3)); ANS(num_cmp(2));
  138 
  139 # $cp->useGrader(~~std_problem_grader);
  140 }
  141 
  142 if($part==2){
  143 BEGIN_TEXT
  144 
  145 Recall the original matrix and your reduced echelon form:
  146 
  147 \[ \hbox{ \[ A = \left[\begin{array}{r r r r r}
  148 $m11 & $m12 & $m13 & $m14 & $m15\cr
  149 $m21 & $m22 & $m23 & $m24 & $m25 \cr
  150 $m31 & $m32 & $m33 & $m34 & $m35 \cr
  151 $m41 & $m42 & $m43 & $m44 & $m45
  152 \end{array} \right]}\quad \text{and} \quad \hbox{\left[\begin{array}{r r r r r}
  153 1 & $a & 0 & $b & $c\cr
  154 0 & 0 & 1 & $d & $d1\cr
  155 0 & 0 & 0 & 0 & 0\cr
  156 0 & 0 & 0 & 0 & 0
  157 \end{array} \right]}\quad .
  158 \]
  159 
  160 $PAR
  161 Give bases for each of the four fundamental subspaces.
  162 $BR
  163 \{mbox('\(\mathbf R(A)\):', ans_array(5,1,8),',', ans_array_extension(5,1,8) ) \}
  164 $BR
  165 \{mbox( '\(\mathbf C(A)\):', ans_array(4,1,8),',', ans_array_extension(4,1,8) ) \}
  166 $BR
  167 \{mbox( '\(\mathbf N(A)\):',  ans_array(5,1,8),',', ans_array_extension(5,1,8), ',', ans_array_extension(5,1,8) ) \}
  168 $BR
  169 \{mbox('\(\mathbf N(A^T)\):', ans_array(4,1,8),',', ans_array_extension(4,1,8) )\}
  170 $BR
  171 
  172 END_TEXT
  173 
  174 ANS(basis_cmp([[1,$a,0,$b,$c],[0,0,1,$d,$d1]]));
  175 ANS(basis_cmp([[$m11,$m21,$m31,$m41],[$m13,$m23,$m33,$m43]]));
  176 ANS(basis_cmp([[-$a,1,0,0,0],[-$b,0,-$d,1,0],[-$c,0,-$d1,0,1]]));
  177 ANS(basis_cmp([[$LN11,$LN12,$LN13,$LN14],[$LN21,$LN22,$LN23,$LN24]]));
  178 
  179 }
  180 
  181 ENDDOCUMENT();  # This should be the last executable line in the problem.