## 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.