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# View of /branches/UGA/4.4.3a.pg

Sat Jul 24 17:09:50 2010 UTC (2 years, 9 months ago) by ted shifrin
File size: 3565 byte(s)
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    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
23            "PGbasicmacros.pl",
24            "PGchoicemacros.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