View of /branches/UGA/5.5.8.pg

    1 ## DESCRIPTION
    2 ## Linear Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('linear algebra','vector space','inner product','dot product','orthogonal','projection')
    6 ## Tagged by cmd6a 4/30/06
    7 
    8 ## DBsubject('Linear Algebra')
    9 ## DBchapter('Vector Spaces')
   10 ## DBsection('Inner Product')
   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 "PGmatrixmacros.pl",
   29 "PGnumericalmacros.pl",
   30 "PGauxiliaryFunctions.pl"
   31 );
   32 
   33 TEXT(beginproblem());
   34 $showPartialCorrectAnswers = 1;
   35 
   36 $a = random(1,9,1);
   37 $b = non_zero_random(-5,5,1);
   38 $c = non_zero_random(-3,3,1);
   39 
   40 $y1 = $b + $a;
   41 $y2 = $b - $a;
   42 $y3 = $c * $b;
   43 
   44 @x = (1, 1, $c);
   45 @y = ($y1, $y2, $y3);
   46 
   47 $norm_x2 = 2+$c*$c;
   48 
   49 @s = NchooseK(3,3);
   50 
   51 # image of (1,0,0):
   52 
   53 $a[0][0] = 1/$norm_x2 + 1/2;
   54 $a[0][1] = 1/$norm_x2 - 1/2;
   55 $a[0][2] = $c/$norm_x2;
   56 
   57 # image of (0,1,0):
   58 
   59 $a[1][0] = 1/$norm_x2 - 1/2;
   60 $a[1][1] = 1/$norm_x2 + 1/2;
   61 $a[1][2] = $c/$norm_x2;
   62 
   63 # image of (0,0,1):
   64 
   65 $a[2][0] = $c/$norm_x2;
   66 $a[2][1] = $c/$norm_x2;
   67 $a[2][2] = $c*$c/$norm_x2;
   68 
   69 BEGIN_TEXT
   70 
   71 Let \(W\) be the subspace of \( {\mathbb R}^3 \) spanned by the vectors
   72 \(\{ display_matrix_mm([[$x[$s[0]]], [$x[$s[1]]], [$x[$s[2]]]])\}\) and
   73 \(\{ display_matrix_mm([[$y[$s[0]]], [$y[$s[1]]], [$y[$s[2]]]])\}\) .
   74 Find the matrix \(A\) of the linear map \(\text{proj}_W\colon\mathbb R^3\to\mathbb R^3\) given by projection onto \(W\).
   75 $BR
   76 \{ mbox( '\(A=\)', answer_matrix(3,3), '.' ) \}
   77 
   78 END_TEXT
   79 
   80 ANS(num_cmp($a[$s[0]][$s[0]]));
   81 ANS(num_cmp($a[$s[1]][$s[0]]));
   82 ANS(num_cmp($a[$s[2]][$s[0]]));
   83 ANS(num_cmp($a[$s[0]][$s[1]]));
   84 ANS(num_cmp($a[$s[1]][$s[1]]));
   85 ANS(num_cmp($a[$s[2]][$s[1]]));
   86 ANS(num_cmp($a[$s[0]][$s[2]]));
   87 ANS(num_cmp($a[$s[1]][$s[2]]));
   88 ANS(num_cmp($a[$s[2]][$s[2]]));
   89 
   90 ENDDOCUMENT();       # This should be the last executable line in the problem.
   91