## DESCRIPTION ## Fundamental Subspaces of a Matrix ## ENDDESCRIPTION ## KEYWORDS('Nullspace', 'Column space', 'Row space', 'Left nullspace') ## ## DBsubject('Calculus') ## DBchapter('') ## DBsection('') ## Date('10/04/2009') ## Author('Ted Shifrin') ## Institution('UGA') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') DOCUMENT(); # This should be the first executable line in the problem. loadMacros("PGstandard.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl", "PGmatrixmacros.pl", "PGmorematrixmacros.pl", "compoundProblem.pl", "PGnumericalmacros.pl", ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = non_zero_random(-3, 3, 1); $b = non_zero_random(-3, 3, 1);$c = non_zero_random(-3, 3, 1); $d = non_zero_random(-3, 3, 1);$d1 = non_zero_random(-3, 3, 1); $e = non_zero_random(-3, 3, 1);$f = non_zero_random(-3, 3, 1); $i = non_zero_random(-1, 1, 2);$g = $e*$f+$i;$k = non_zero_random(-3, 3, 1); $l = non_zero_random(-3, 3, 1);$m = non_zero_random(-3, 3, 1); $n = non_zero_random(-3, 3, 1);$m11=1; $m12=$a; $m13=$e; $m14=$b+$e*$d; $m15=$c+$e*$d1; $m21=$f; $m22=$f*$a;$m23=$g;$m24=$f*$b+$g*$d; $m25=$f*$c+$g*$d1;$m31=$k;$m32=$k*$a; $m33=$l; $m34=$k*$b+$l*$d;$m35=$k*$c+$l*$d1; $m41=$m; $m42=$m*$a;$m43=$n;$m44=$m*$b+$n*$d; $m45=$m*$c+$n*$d1;$LN11=-$k+$i*$f*($l-$e*$k); $LN12=-$i*($l-$e*$k);$LN13=1; $LN14=0;$LN21=-$m+$i*$f*($n-$e*$m); $LN22=-$i*($n-$e*$m);$LN23=0; $LN24=1;$isProfessor = ($studentLogin eq 'shifrin' ||$studentLogin eq 'test'); $cp = new compoundProblem( parts=>2, weights=>[.4,.6], parserValues=>1, allowReset =>$isProfessor, nextVisible => 'Always', nextStyle => 'Button', ); $part =$cp->part; if($part==1){ BEGIN_TEXT Consider the matrix $A = \left[\begin{array}{r r r r r} m11 & m12 & m13 & m14 & m15\cr m21 & m22 & m23 & m24 & m25 \cr m31 & m32 & m33 & m34 & m35 \cr m41 & m42 & m43 & m44 & m45 \end{array} \right]\quad .$$PAR We start with some basic facts about the four fundamental subspaces associated to $$A$$: $PAR $$\mathbf R(A)$$ is a subspace of $$\mathbb R^k$$ with $$k =$$ \{ans_rule(3)\}.$BR $$\mathbf C(A)$$ is a subspace of $$\mathbb R^k$$ with $$k =$$ \{ans_rule(3)\}. $BR $$\mathbf N(A)$$ is a subspace of $$\mathbb R^k$$ with $$k =$$ \{ans_rule(3)\}.$BR $$\mathbf N(A^T)$$ is a subspace of $$\mathbb R^k$$ with $$k =$$ \{ans_rule(3)\}. $PAR Give the reduced echelon form of $$A$$.$BR \{ mbox( answer_matrix(4,5,5) )\} $PAR Give the dimension of each of the fundamental subspaces:$BR dim $$\mathbf R(A) =$$ \{ ans_rule(5) \} $BR dim $$\mathbf C(A) =$$ \{ ans_rule(5) \}$BR dim $$\mathbf N(A) =$$ \{ ans_rule(5) \} $BR dim $$\mathbf N(A^T) =$$ \{ ans_rule(5) \}$PAR When you have answered all these questions correctly, click on the "Go on to next part" button. END_TEXT ANS(num_cmp(5)); ANS(num_cmp(4)); ANS(num_cmp(5)); ANS(num_cmp(4)); ANS(num_cmp(1)); ANS(num_cmp($a)); ANS(num_cmp(0)); ANS(num_cmp($b)); ANS(num_cmp($c)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(1)); ANS(num_cmp($d)); ANS(num_cmp($d1)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(2)); ANS(num_cmp(2)); ANS(num_cmp(3)); ANS(num_cmp(2)); #$cp->useGrader(~~std_problem_grader); } if($part==2){ BEGIN_TEXT Recall the original matrix and your reduced echelon form: $\hbox{ \[ A = \left[\begin{array}{r r r r r} m11 & m12 & m13 & m14 & m15\cr m21 & m22 & m23 & m24 & m25 \cr m31 & m32 & m33 & m34 & m35 \cr m41 & m42 & m43 & m44 & m45 \end{array} \right]}\quad \text{and} \quad \hbox{\left[\begin{array}{r r r r r} 1 & a & 0 & b & c\cr 0 & 0 & 1 & d & d1\cr 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 \end{array} \right]}\quad .$$PAR Give bases for each of the four fundamental subspaces. $BR \{mbox('$$\mathbf R(A)$$:', ans_array(5,1,8),',', ans_array_extension(5,1,8) ) \}$BR \{mbox( '$$\mathbf C(A)$$:', ans_array(4,1,8),',', ans_array_extension(4,1,8) ) \} $BR \{mbox( '$$\mathbf N(A)$$:', ans_array(5,1,8),',', ans_array_extension(5,1,8), ',', ans_array_extension(5,1,8) ) \}$BR \{mbox('$$\mathbf N(A^T)$$:', ans_array(4,1,8),',', ans_array_extension(4,1,8) )\} $BR END_TEXT ANS(basis_cmp([[1,$a,0,$b,$c],[0,0,1,$d,$d1]])); ANS(basis_cmp([[$m11,$m21,$m31,$m41],[$m13,$m23,$m33,$m43]])); ANS(basis_cmp([[-$a,1,0,0,0],[-$b,0,-$d,1,0],[-$c,0,-$d1,0,1]])); ANS(basis_cmp([[$LN11,$LN12,$LN13,$LN14],[$LN21,$LN22,$LN23,\$LN24]])); } ENDDOCUMENT(); # This should be the last executable line in the problem.