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Sat Jul 24 17:09:50 2010 UTC (2 years, 10 months ago) by ted shifrin
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    1 ## DESCRIPTION
2 ## Linear Algebra
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('linear algebra','vector space','basis','dot product','inner product','orthonormal')
6 ## Tagged by cmd6a 5/3/06
7
8 ## DBsubject('Linear Algebra')
9 ## DBchapter('Vector Spaces')
10 ## DBsection('Bases')
11 ## Date('')
12 ## Author('modified by Shifrin')
13 ## Institution('Rochester')
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(
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.pl",
26 "PGanswermacros.pl",
27 "PGgraphmacros.pl",
28 "PGnumericalmacros.pl",
29 "PGstatisticsmacros.pl",
30 "PGauxiliaryFunctions.pl",
31 "PGmatrixmacros.pl"
32 );
33
34 TEXT(beginproblem());
35 $showPartialCorrectAnswers = 1; 36 37$a = non_zero_random(-4,4,1);
38 $b = non_zero_random(-4,4,1); 39$c = non_zero_random(-3,3,1);
40 $d = non_zero_random(-4,4,1); 41 42 # create orthogonal vectors 43 44$x[1] = $a; 45$x[2] = $c; 46$x[3] = $b; 47$x[4] = 0;
48
49 $y[1] =$b;
50 $y[2] = 0; 51$y[3] = - $a; 52$y[4] = $d; 53 54 #change the vectors so that they are no longer orthogonal but the Gram-Schmidt precess gives the original x,y. 55 56$p = random(1,3,1);
57 $q = random(1,3,1); 58$k = non_zero_random(-2,2,1);
59
60 $norm_x = 0; 61$norm_y = 0;
62
63 for ($i=1;$i<5; $i++) { 64$cx[$i] =$p * $x[$i];
65   $cy[$i] = $q *$y[$i] +$k * $x[$i];
66         $norm_x =$norm_x + $x[$i] * $x[$i];
67         $norm_y =$norm_y + $y[$i] * $y[$i];
68 }
69
70 $norm_x = sqrt($norm_x);
71 $norm_y = sqrt($norm_y);
72
73 BEGIN_TEXT
74
75 Let $$\mathbf v = \{display_matrix_mm([[cx[1]], [cx[2]], [cx[3]], [cx[4]]])\}$$ and $$\mathbf w = \{ display_matrix_mm([[cy[1]], [cy[2]], [cy[3]], [cy[4]]])\}$$ .
76 $BR 77 Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of $$\mathbb R^4$$ 78 spanned by $$\mathbf v$$ and $$\mathbf w$$. 79 80 \{ mbox( answer_matrix(4,1,10), ', ', answer_matrix(4,1,10), '.' ) \} 81 82 END_TEXT 83 84 ANS(num_cmp($x[1]/$norm_x)); 85 ANS(num_cmp($x[2]/$norm_x)); 86 ANS(num_cmp($x[3]/$norm_x)); 87 ANS(num_cmp($x[4]/$norm_x)); 88 89 ANS(num_cmp($y[1]/$norm_y)); 90 ANS(num_cmp($y[2]/$norm_y)); 91 ANS(num_cmp($y[3]/$norm_y)); 92 ANS(num_cmp($y[4]/\$norm_y));
93
94 ENDDOCUMENT();       # This should be the last executable line in the problem.
95


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