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    1 ## DESCRIPTION
2 ## Linear Algebra
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS ('linear algebra','vector space','inner product','dot product')
6 ## Tagged by cmd6a 4/30/06
7
8 ## DBsubject('Linear Algebra')
9 ## DBchapter('Vector Spaces')
10 ## DBsection('Inner Product')
11 ## Date('')
12 ## Author('')
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
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.pl",
27 "PGgraphmacros.pl",
28 "PGmatrixmacros.pl",
29 "PGnumericalmacros.pl",
30 "PGauxiliaryFunctions.pl"
31 );
32
33 TEXT(beginproblem());
34 $showPartialCorrectAnswers = 1; 35 36$A = new Matrix(3,2);
37 $B = new Matrix(3,2); 38 39 foreach my$i (1..3) {
40         foreach my $j (1..2) { 41$A->assign($i,$j, non_zero_random(-3,3,1));
42     $B->assign($i,$j, non_zero_random(-3,3,1)); 43 } 44 } 45 46$AT = new Matrix(2,3);
47 $BT = new Matrix(2,3); 48 49$AT->transpose($A); 50$BT->transpose($B); 51 52$AB = $AT *$B;
53 $prod =$AB->element(1,1) + $AB->element(2,2); 54 55$A2 = $AT *$A;
56 $norm_a = sqrt($A2->element(1,1) + $A2->element(2,2)); 57 58$B2 = $BT *$B;
59 $norm_b = sqrt($B2->element(1,1) + $B2->element(2,2)); 60 61$alpha = arccos($prod /$norm_a / $norm_b); 62 63 BEGIN_TEXT 64 65 If $$A$$ and $$B$$ are arbitrary $$m\times n$$ matrices, then the mapping $$<A,B>= {\rm trace}(A^T B)$$ 66 defines an inner product in $${\mathbb R}^{m\times n}$$. 67$BR
68 Use this inner product to find $$<A,B>$$, the norms
69 $$||A||$$ and $$||B||$$, and the angle $$\alpha_{A,B}$$ between $$A$$ and $$B$$ for
70 $BR 71 \{ mbox( '$$A=$$', display_matrix($A), 'and $$B=$$', display_matrix($B), '.' ) \} 72$BR
73 $$<A,B>=$$ \{ans_rule(20)\},
74 $BR 75 $$||A||=$$ \{ans_rule(20)\}, 76$BR
77 $$||B||=$$ \{ans_rule(20)\},
78 $BR 79 $$\alpha_{A,B}=$$ \{ans_rule(20)\}. 80 81 END_TEXT 82 83 ANS(num_cmp($prod));
84 ANS(num_cmp($norm_a)); 85 ANS(num_cmp($norm_b));
86 ANS(num_cmp(\$alpha));
87
88 ENDDOCUMENT();       # This should be the last executable line in the problem.
89