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| 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 : |
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