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# View of /trunk/NationalProblemLibrary/Rochester/setLinearAlgebra15TransfOfLinSpaces/ur_la_15_15.pg

Thu May 4 16:40:05 2006 UTC (7 years ago) by jjholt
File size: 1957 byte(s)
Added tags.  --JH


    1 ## DESCRIPTION
2 ## Linear Algebra
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS ('linear algebra','vector space','linear transformation')
6 ## Tagged by cmd6a 5/3/06
7
8 ## DBsubject('Linear Algebra')
9 ## DBchapter('Vector Spaces')
10 ## DBsection('Linear Transformations')
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 = random(2,4,1);
37 $b = random(2,4,1); 38$sa = random(-1,1,2);
39 $sb = random(-1,1,2); 40$a = $a *$sa;
41 $b =$b * $sb; 42 43$v2 = random(2,4,1);
44 $v3 = random(-2,-4,1); 45$v1 = - $a*$v2 - $b*$v3;
46
47 $v = new Matrix(3,2); 48$v->assign(1,1, - $a); 49$v->assign(2,1,1);
50 $v->assign(3,1,0); 51$v->assign(1,2,- $b); 52$v->assign(2,2,0);
53 $v->assign(3,2,1); 54 55$C = new Matrix(2,3);
56 foreach $i (1..2) { 57 foreach$j (1..3){
58                 $C->assign($i, $j, non_zero_random(-2,2,1)); 59 } 60 } 61 62$M = new Matrix(3,3);
63 $M =$v * $C; 64 65$N = new Matrix(3,2);
66 $N =$M * $v; 67 68$ans1 = $N->element(2,1); 69$ans2 = $N->element(2,2); 70$ans3 = $N->element(3,1); 71$ans4 = $N->element(3,2); 72 73$RIGHT_BRACE = '\}';
74
75 BEGIN_TEXT
76
77 Let $$V$$ be the plane with equation $$x_1 + a x_2 + b x_3 =0$$ in $${\mathbb R}^3$$.
78 Find the matrix $$A$$ of the linear transformation
79 $$T(x)= \{display_matrix_mm(M)\} x$$ with respect to the basis
80 $$\left\{'\{'\} \{display_matrix_mm([[- a], [1], [0]])\} , \{display_matrix_mm([[- b], [0], [1]])\} 81 \right{RIGHT_BRACE}$$.
82 $BR 83 \{ mbox( '$$A=$$', answer_matrix(2,2,10) ) \} 84 85 END_TEXT 86 87 ANS(num_cmp($ans1));
88 ANS(num_cmp($ans2)); 89 ANS(num_cmp($ans3));
90 ANS(num_cmp(\$ans4));
91
92 ENDDOCUMENT();       # This should be the last executable line in the problem.
93


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