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

 1 : jjholt 200 ## DESCRIPTION 2 : ## Linear Algebra 3 : ## ENDDESCRIPTION 4 : jj 144 5 : jjholt 200 ## 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 : jj 144 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(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 :