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| 1 : | ted shifri | 1457 | ## 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. |
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