View of /branches/UGA/5.5.11.pg

    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