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| 1 : | ted shifri | 1457 | ## DESCRIPTION |
| 2 : | ## Linear Algebra | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | ## KEYWORDS('Quadratic form' 'Decomposition of Symmetric Matrix') | ||
| 6 : | ## Tagged by tda2d | ||
| 7 : | |||
| 8 : | ## DBsubject('Linear Algebra') | ||
| 9 : | ## DBchapter('Matrices') | ||
| 10 : | ## DBsection('Quadratic Forms') | ||
| 11 : | ## Date('') | ||
| 12 : | ## Author('Shifrin') | ||
| 13 : | ## Institution('UGA') | ||
| 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 : | "PGauxiliaryFunctions.pl", | ||
| 28 : | "PGgraphmacros.pl", | ||
| 29 : | "PGmatrixmacros.pl", | ||
| 30 : | # "PGmorematrixmacros.pl" | ||
| 31 : | "weightedGrader.pl" | ||
| 32 : | ); | ||
| 33 : | |||
| 34 : | |||
| 35 : | |||
| 36 : | TEXT(beginproblem()); | ||
| 37 : | install_weighted_grader(); | ||
| 38 : | $showPartialCorrectAnswers = 1; | ||
| 39 : | |||
| 40 : | $L = new Matrix(2,2); | ||
| 41 : | $D = new Matrix(2,2); | ||
| 42 : | $LT = new Matrix(2,2); | ||
| 43 : | $A = new Matrix(2,2); | ||
| 44 : | |||
| 45 : | $d11 = non_zero_random(-5,5,1); | ||
| 46 : | $d22 = random(-5,5,1); | ||
| 47 : | $l21 = random(-5,5,1); | ||
| 48 : | |||
| 49 : | $D->assign(1,1,$d11); | ||
| 50 : | $D->assign(2,2,$d22); | ||
| 51 : | $L->assign(2,1,$l21); | ||
| 52 : | $LT->assign(1,2,$l21); | ||
| 53 : | |||
| 54 : | foreach $i (1..2){ | ||
| 55 : | foreach $j (1..2){ | ||
| 56 : | if($j!=$i) { | ||
| 57 : | $D->assign($i,$j,0);}}} | ||
| 58 : | |||
| 59 : | foreach $i (1..2){ | ||
| 60 : | foreach $j (1..2){ | ||
| 61 : | if($j==$i) { | ||
| 62 : | $L->assign($i,$j,1); $LT->assign($i,$j,1);} elsif($j>$i){ | ||
| 63 : | $L->assign($i,$j,0); $LT->assign($j,$i,0);}}} | ||
| 64 : | |||
| 65 : | $A = $L*$D*$LT; | ||
| 66 : | |||
| 67 : | $ques=("The quadratic form \( \mathcal Q(\mathbf x) = \mathbf x^T A \mathbf x \) is"); | ||
| 68 : | |||
| 69 : | if($d11>0 and $d22>0){$ans="positive definite";} | ||
| 70 : | elsif(($d11==0 and $d22>=0) or ($d11>0 and $d22==0)){$ans="positive semidefinite";} | ||
| 71 : | elsif($d11<0 and $d22<0){$ans="negative definite";} | ||
| 72 : | elsif(($d11==0 and $d22 le 0) or ($d11<0 and $d22==0)){$ans="negative semidefinite";} | ||
| 73 : | else {$ans="indefinite";} | ||
| 74 : | |||
| 75 : | $mc0 = new_multiple_choice(); | ||
| 76 : | $mc0->qa($ques, $ans); | ||
| 77 : | $mc0->makeLast("positive definite","negative definite","positive semidefinite", "negative semidefinite","indefinite"); | ||
| 78 : | |||
| 79 : | BEGIN_TEXT | ||
| 80 : | |||
| 81 : | If \(A=\{display_matrix_mm($A)\}\), | ||
| 82 : | $BR | ||
| 83 : | then we can write \( A = LDL^T \), where | ||
| 84 : | $PAR | ||
| 85 : | \{mbox( '\(L = \)', display_matrix([['\(1\)','\(0\)'],[ans_rule(5),'\(1\)']]) )\} | ||
| 86 : | $BR | ||
| 87 : | \{mbox('and', '\( D = \)', display_matrix([[ans_rule(5),'\(0\)'],['\(0\)',ans_rule(5)]]), '.')\} | ||
| 88 : | $PAR | ||
| 89 : | |||
| 90 : | \{ $mc0->print_q() \} | ||
| 91 : | \{ $mc0->print_a() \} | ||
| 92 : | |||
| 93 : | END_TEXT | ||
| 94 : | |||
| 95 : | WEIGHTED_ANS(num_cmp($l21),25, | ||
| 96 : | num_cmp($d11),25, | ||
| 97 : | num_cmp($d22),25, | ||
| 98 : | radio_cmp($mc0->correct_ans),25); | ||
| 99 : | |||
| 100 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 101 : | |||
| 102 : | ANS(num_cmp ra_flatten_matrix($C*$B)); |
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