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

    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 
   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