View of /trunk/NationalProblemLibrary/Rochester/setLinearAlgebra15TransfOfLinSpaces/ur_la_15_4.pg

    1 ##DESCRIPTION
    2 ##KEYWORDS('linear albebra', 'vector space', 'linear transformation', 'matrix', 'basis')
    3 ##
    4 ##ENDDESCRIPTION
    5 
    6 DOCUMENT();        # This should be the first executable line in the problem.
    7 
    8 loadMacros(
    9 "PG.pl",
   10 "PGbasicmacros.pl",
   11 "PGchoicemacros.pl",
   12 "PGanswermacros.pl",
   13 "PGgraphmacros.pl",
   14 "PGnumericalmacros.pl",
   15 "PGstatisticsmacros.pl",
   16 "PGmatrixmacros.pl"
   17 );
   18 
   19 TEXT(beginproblem());
   20 $showPartialCorrectAnswers = 1;
   21 
   22 $a = random(2,15,1);
   23 $b = random(2,15,1);
   24 while ($a == $b) { $b = random(2,15,1); }
   25 
   26 BEGIN_TEXT
   27 
   28 \{ mbox( 'The matrices \(A_1=\)', display_matrix([[1, 0], [0, 0]]), ', ',
   29    ' \( A_2=\)', display_matrix([[0, 1], [0, 0]]), ',' ) \}
   30 $BR
   31 \{ mbox( '\(A_3=\)', display_matrix([[0, 0], [1, 0]]), ', ',
   32    ' and \(A_4=\)', display_matrix([[0, 0], [0, 1]]) ) \}
   33 $BR
   34 form a basis for the linear space \(V={\mathbb R}^{2\times 2}.\)
   35 Write the matrix of the linear transformation \(T:{\mathbb R}^{2\times 2} \rightarrow {\mathbb R}^{2\times 2}\)
   36 such that \(T(A)=$a A + $b A^T\) relative to this
   37 basis:
   38 $BR
   39 \{ answer_matrix(4,4,5) \}
   40 
   41 END_TEXT
   42 
   43 $c = $a+$b;
   44 
   45 ANS(num_cmp($c));
   46 ANS(num_cmp(0));
   47 ANS(num_cmp(0));
   48 ANS(num_cmp(0));
   49 ANS(num_cmp(0));
   50 ANS(num_cmp($a));
   51 ANS(num_cmp($b));
   52 ANS(num_cmp(0));
   53 ANS(num_cmp(0));
   54 ANS(num_cmp($b));
   55 ANS(num_cmp($a));
   56 ANS(num_cmp(0));
   57 ANS(num_cmp(0));
   58 ANS(num_cmp(0));
   59 ANS(num_cmp(0));
   60 ANS(num_cmp($a+$b));
   61 
   62 
   63 ENDDOCUMENT();       # This should be the last executable line in the problem.
   64