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

Sun Mar 26 15:53:04 2006 UTC (7 years, 1 month ago) by jj
File size: 1666 byte(s)
Initial import


    1 ##DESCRIPTION
2 ##KEYWORDS('linear', 'transformation', 'plane', 'basis', 'matrix')
3 ##
4 ##ENDDESCRIPTION
5
6 DOCUMENT();        # This should be the first executable line in the problem.
7
9 "PG.pl",
10 "PGbasicmacros.pl",
11 "PGchoicemacros.pl",
13 "PGgraphmacros.pl",
14 "PGmatrixmacros.pl",
15 "PGnumericalmacros.pl",
16 "PGauxiliaryFunctions.pl"
17 );
18
19 TEXT(beginproblem());
20 $showPartialCorrectAnswers = 1; 21 22$a = random(2,4,1);
23 $b = random(2,4,1); 24$sa = random(-1,1,2);
25 $sb = random(-1,1,2); 26$a = $a *$sa;
27 $b =$b * $sb; 28 29$v2 = random(2,4,1);
30 $v3 = random(-2,-4,1); 31$v1 = - $a*$v2 - $b*$v3;
32
33 $v = new Matrix(3,2); 34$v->assign(1,1, - $a); 35$v->assign(2,1,1);
36 $v->assign(3,1,0); 37$v->assign(1,2,- $b); 38$v->assign(2,2,0);
39 $v->assign(3,2,1); 40 41$C = new Matrix(2,3);
42 foreach $i (1..2) { 43 foreach$j (1..3){
44                 $C->assign($i, $j, non_zero_random(-2,2,1)); 45 } 46 } 47 48$M = new Matrix(3,3);
49 $M =$v * $C; 50 51$N = new Matrix(3,2);
52 $N =$M * $v; 53 54$ans1 = $N->element(2,1); 55$ans2 = $N->element(2,2); 56$ans3 = $N->element(3,1); 57$ans4 = $N->element(3,2); 58 59$RIGHT_BRACE = '\}';
60
61 BEGIN_TEXT
62
63 Let $$V$$ be the plane with equation $$x_1 + a x_2 + b x_3 =0$$ in $${\mathbb R}^3$$.
64 Find the matrix $$A$$ of the linear transformation
65 $$T(x)= \{display_matrix_mm(M)\} x$$ with respect to the basis
66 $$\left\{'\{'\} \{display_matrix_mm([[- a], [1], [0]])\} , \{display_matrix_mm([[- b], [0], [1]])\} 67 \right{RIGHT_BRACE}$$.
68 $BR 69 \{ mbox( '$$A=$$', answer_matrix(2,2,10) ) \} 70 71 END_TEXT 72 73 ANS(num_cmp($ans1));
74 ANS(num_cmp($ans2)); 75 ANS(num_cmp($ans3));
76 ANS(num_cmp(\$ans4));
77
78 ENDDOCUMENT();       # This should be the last executable line in the problem.
79