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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','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


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