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| 1 : | ted shifri | 1458 | ## DESCRIPTION |
| 2 : | ## Linear Algebra | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | ## KEYWORDS ('linear algebra','vector space','linear transformation') | ||
| 6 : | |||
| 7 : | ## DBsubject('Multivariable Mathematics') | ||
| 8 : | ## DBchapter('Linear Transformations') | ||
| 9 : | ## DBsection('Change of Basis') | ||
| 10 : | ## Date('') | ||
| 11 : | ## Author('Shifrin') | ||
| 12 : | ## Institution('UGA') | ||
| 13 : | ## TitleText1('') | ||
| 14 : | ## EditionText1('') | ||
| 15 : | ## AuthorText1('') | ||
| 16 : | ## Section1('') | ||
| 17 : | ## Problem1('') | ||
| 18 : | |||
| 19 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 20 : | |||
| 21 : | loadMacros( | ||
| 22 : | "PG.pl", | ||
| 23 : | "PGbasicmacros.pl", | ||
| 24 : | "PGchoicemacros.pl", | ||
| 25 : | "PGanswermacros.pl", | ||
| 26 : | "PGmatrixmacros.pl", | ||
| 27 : | "PGmorematrixmacros.pl", | ||
| 28 : | "weightedGrader.pl", | ||
| 29 : | "MathObjects.pl", | ||
| 30 : | "PGauxiliaryFunctions.pl", | ||
| 31 : | ); | ||
| 32 : | |||
| 33 : | |||
| 34 : | TEXT(beginproblem()); | ||
| 35 : | $showPartialCorrectAnswers = 1; | ||
| 36 : | |||
| 37 : | install_weighted_grader(); | ||
| 38 : | |||
| 39 : | Context("Numeric")->variables->are(x=>'Real', y=>'Real', z=>'Real'); | ||
| 40 : | |||
| 41 : | @normal=("1","2","2"); | ||
| 42 : | @choice=NchooseK(3,3); | ||
| 43 : | $sgn[1] = 1; | ||
| 44 : | $sgn[2] = random(-1,1,2); | ||
| 45 : | $sgn[3] = random(-1,1,2); | ||
| 46 : | foreach $i(1..3) {$a[$i]=$sgn[$i]*@normal[@choice[$i-1]]}; | ||
| 47 : | |||
| 48 : | $g=Formula("$a[1]x+$a[2]y+$a[3]z")->reduce; | ||
| 49 : | |||
| 50 : | $M = new Matrix(3,3); | ||
| 51 : | $R = new Matrix(3,3); | ||
| 52 : | $M -> assign(1,1,1); $R -> assign(1,1,1); | ||
| 53 : | $M -> assign(2,2,1); $R -> assign(2,2,1); | ||
| 54 : | $M -> assign(3,3,0); $R -> assign(3,3,-1); | ||
| 55 : | foreach $i(1..3) {foreach $j(1..3) {if($i!=$j) {$M -> assign($i,$j,0); | ||
| 56 : | $R -> assign($i,$j,0);}}} | ||
| 57 : | # $M -> assign(1,2,0); $M -> assign(2,1,0); $M -> assign(1,3,0); $M -> | ||
| 58 : | # assign(3,1,0); $M -> assign(2,3,0); $M -> assign(3,2,0); | ||
| 59 : | |||
| 60 : | $T = new Matrix(3,3); | ||
| 61 : | $S = new Matrix(3,3); | ||
| 62 : | $denom = $a[1]**2+$a[2]**2+$a[3]**2; | ||
| 63 : | # $T = Matrix("[[($a[2]**2+$a[3]**2)/$denom,-$a[1]*$a[2]/$denom,-$a[1]*$a[3] # /$denom],[-$a[1]*$a[2]/$denom,($a[1]**2+$a[3]**2)/$denom,-$a[2]*$a[3]/$denom], | ||
| 64 : | # [-$a[1]*$a[3]/$denom,-$a[2]*$a[3]/$denom,($a[1]**2+$a[2]**2)/$denom]]"); | ||
| 65 : | $T -> assign(1,1,($a[2]**2+$a[3]**2)/$denom); | ||
| 66 : | $T -> assign(1,2,-$a[1]*$a[2]/$denom); $T -> assign(2,1,$T->element(1,2)); | ||
| 67 : | $T -> assign(1,3,-$a[1]*$a[3]/$denom); $T -> assign(3,1,$T->element(1,3)); | ||
| 68 : | $T -> assign(2,2,($a[1]**2+$a[3]**2)/$denom); | ||
| 69 : | $T -> assign(2,3,-$a[2]*$a[3]/$denom); $T -> assign(3,2,$T->element(2,3)); | ||
| 70 : | $T -> assign(3,3,($a[1]**2+$a[2]**2)/$denom); | ||
| 71 : | |||
| 72 : | $S -> assign(1,1,(-$a[1]**2+$a[2]**2+$a[3]**2)/$denom); | ||
| 73 : | $S -> assign(1,2,-2*$a[1]*$a[2]/$denom); $S -> assign(2,1,-2*$a[1]*$a[2]/$denom); | ||
| 74 : | $S -> assign(1,3,-2*$a[1]*$a[3]/$denom); $S -> assign(3,1,-2*$a[1]*$a[3]/$denom); | ||
| 75 : | $S -> assign(2,2,($a[1]**2-$a[2]**2+$a[3]**2)/$denom); | ||
| 76 : | $S -> assign(2,3,-2*$a[2]*$a[3]/$denom); $S -> assign(3,2,-2*$a[2]*$a[3]/$denom); | ||
| 77 : | $S -> assign(3,3,($a[1]**2+$a[2]**2-$a[3]**2)/$denom); | ||
| 78 : | |||
| 79 : | Context()->texStrings; | ||
| 80 : | BEGIN_TEXT | ||
| 81 : | |||
| 82 : | Consider the \(2\)-dimensional subspace \(\mathcal P\subset\mathbb R^3\) defined by | ||
| 83 : | \[ $g = 0 .\] | ||
| 84 : | $PAR | ||
| 85 : | (a) Give a basis \(\mathcal B = \lbrace \mathbf v_1,\mathbf v_2,\mathbf v_3 \rbrace \) so that \(\mathbf v_1,\mathbf v_2\) span the plane \(\mathcal P\) and \(\mathbf v_3\) is orthogonal to it. | ||
| 86 : | $BR | ||
| 87 : | \{mbox('\(\mathbf v_1 = \)', ans_array(3,1,8), ', \(\quad \mathbf v_2 = \)', ans_array_extension(3,1,8), ', \(\quad \mathbf v_3 = \)', ans_array(3,1,8), '.') \} | ||
| 88 : | $PAR | ||
| 89 : | (b) Let \(T\colon \mathbb R^3\to\mathbb R^3\) be the linear transformation defined by projecting onto the plane. Give the matrix representing \(T\) with respect to the basis \(\mathcal B\). | ||
| 90 : | $BR | ||
| 91 : | \{mbox('\([T]_{\mathcal B} = \)', answer_matrix(3,3,3), '.')\} | ||
| 92 : | $PAR | ||
| 93 : | (c) Use the change-of-basis formula to calculate the standard matrix of \(T\): | ||
| 94 : | $BR | ||
| 95 : | \{mbox ( '\([T] = \)', answer_matrix(3,3,8), '.') \} | ||
| 96 : | (d) Now let \(S\colon \mathbb R^3\to\mathbb R^3\) be the linear transformation defined by reflecting across the plane \(\mathcal P\). Give the matrix representing \(S\) with respect to the basis \(\mathcal B\). | ||
| 97 : | $BR | ||
| 98 : | \{mbox('\([S]_{\mathcal B} = \)', answer_matrix(3,3,3), '.')\} | ||
| 99 : | $PAR | ||
| 100 : | (e) Use the change-of-basis formula to calculate the standard matrix of \(S\): | ||
| 101 : | $BR | ||
| 102 : | \{mbox ( '\([S] = \)', answer_matrix(3,3,8), '.') \} | ||
| 103 : | |||
| 104 : | END_TEXT | ||
| 105 : | |||
| 106 : | Context("Vector"); | ||
| 107 : | WEIGHTED_ANS(basis_cmp([[$a[3],0,-$a[1]],[0,$a[3],-$a[2]]]),7); | ||
| 108 : | WEIGHTED_ANS(basis_cmp([[$a[1],$a[2],$a[3]]]),3); | ||
| 109 : | WEIGHTED_ANS(num_cmp($M->element(1,1)),1,num_cmp($M->element(1,2)),1,num_cmp($M->element(1,3)),1,num_cmp($M->element(2,1)),1,num_cmp($M->element(2,2)),1,num_cmp($M->element(2,3)),1,num_cmp($M->element(3,1)),1,num_cmp($M->element(3,2)),1,num_cmp($M->element(3,3)),1); | ||
| 110 : | WEIGHTED_ANS(num_cmp($T->element(1,1)),4,num_cmp($T->element(1,2)),4,num_cmp($T->element(1,3)),4,num_cmp($T->element(2,1)),4,num_cmp($T->element(2,2)),4,num_cmp($T->element(2,3)),4,num_cmp($T->element(3,1)),4,num_cmp($T->element(3,2)),4,num_cmp($T->element(3,3)),4); | ||
| 111 : | WEIGHTED_ANS(num_cmp($R->element(1,1)),1,num_cmp($R->element(1,2)),1,num_cmp($R->element(1,3)),1,num_cmp($R->element(2,1)),1,num_cmp($R->element(2,2)),1,num_cmp($R->element(2,3)),1,num_cmp($R->element(3,1)),1,num_cmp($R->element(3,2)),1,num_cmp($R->element(3,3)),1,); | ||
| 112 : | WEIGHTED_ANS(num_cmp($S->element(1,1)),4,num_cmp($S->element(1,2)),4,num_cmp($S->element(1,3)),4,num_cmp($S->element(2,1)),4,num_cmp($S->element(2,2)),4,num_cmp($S->element(2,3)),4,num_cmp($S->element(3,1)),4,num_cmp($S->element(3,2)),4,num_cmp($S->element(3,3)),4); | ||
| 113 : | |||
| 114 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 115 : |
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