## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## KEYWORDS ('linear algebra','vector space','linear transformation')

## DBsubject('Multivariable Mathematics')
## DBchapter('Linear Transformations')
## DBsection('Change of Basis')
## Date('')
## Author('Shifrin')
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGmatrixmacros.pl",
"PGmorematrixmacros.pl",
"weightedGrader.pl",
"MathObjects.pl",
"PGauxiliaryFunctions.pl",
);


TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

install_weighted_grader();

Context("Numeric")->variables->are(x=>'Real', y=>'Real', z=>'Real');

@normal=("1","2","2");
@choice=NchooseK(3,3);
$sgn[1] = 1;
$sgn[2] = random(-1,1,2);
$sgn[3] = random(-1,1,2);
foreach $i(1..3) {$a[$i]=$sgn[$i]*@normal[@choice[$i-1]]};

$g=Formula("$a[1]x+$a[2]y+$a[3]z")->reduce;

$M = new Matrix(3,3);
$R = new Matrix(3,3);
$M -> assign(1,1,1); $R -> assign(1,1,1);
$M -> assign(2,2,1); $R -> assign(2,2,1);
$M -> assign(3,3,0); $R -> assign(3,3,-1);
foreach $i(1..3) {foreach $j(1..3) {if($i!=$j) {$M -> assign($i,$j,0);
$R -> assign($i,$j,0);}}}
# $M -> assign(1,2,0); $M -> assign(2,1,0); $M -> assign(1,3,0); $M -> 
# assign(3,1,0); $M -> assign(2,3,0); $M -> assign(3,2,0);

$T = new Matrix(3,3);
$S = new Matrix(3,3);
$denom = $a[1]**2+$a[2]**2+$a[3]**2;
# $T = Matrix("[[($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]/$denom],
# [-$a[1]*$a[3]/$denom,-$a[2]*$a[3]/$denom,($a[1]**2+$a[2]**2)/$denom]]");
$T -> assign(1,1,($a[2]**2+$a[3]**2)/$denom);
$T -> assign(1,2,-$a[1]*$a[2]/$denom); $T -> assign(2,1,$T->element(1,2));
$T -> assign(1,3,-$a[1]*$a[3]/$denom); $T -> assign(3,1,$T->element(1,3));
$T -> assign(2,2,($a[1]**2+$a[3]**2)/$denom);
$T -> assign(2,3,-$a[2]*$a[3]/$denom); $T -> assign(3,2,$T->element(2,3));
$T -> assign(3,3,($a[1]**2+$a[2]**2)/$denom);

$S -> assign(1,1,(-$a[1]**2+$a[2]**2+$a[3]**2)/$denom);
$S -> assign(1,2,-2*$a[1]*$a[2]/$denom); $S -> assign(2,1,-2*$a[1]*$a[2]/$denom);
$S -> assign(1,3,-2*$a[1]*$a[3]/$denom); $S -> assign(3,1,-2*$a[1]*$a[3]/$denom);
$S -> assign(2,2,($a[1]**2-$a[2]**2+$a[3]**2)/$denom);
$S -> assign(2,3,-2*$a[2]*$a[3]/$denom); $S -> assign(3,2,-2*$a[2]*$a[3]/$denom);
$S -> assign(3,3,($a[1]**2+$a[2]**2-$a[3]**2)/$denom);

Context()->texStrings;
BEGIN_TEXT

Consider the \(2\)-dimensional subspace \(\mathcal P\subset\mathbb R^3\) defined by
\[ $g = 0 .\]
$PAR
(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.
$BR
\{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), '.') \}
$PAR
(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 \(\mathcal B\).
$BR
\{mbox('\([T]_{\mathcal B} = \)', answer_matrix(3,3,3), '.')\}
$PAR
(c) Use the change-of-basis formula to calculate the standard matrix of \(T\):
$BR
\{mbox ( '\([T] = \)', answer_matrix(3,3,8), '.') \}
(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 \(\mathcal B\).
$BR
\{mbox('\([S]_{\mathcal B} = \)', answer_matrix(3,3,3), '.')\}
$PAR
(e) Use the change-of-basis formula to calculate the standard matrix of \(S\):
$BR
\{mbox ( '\([S] = \)', answer_matrix(3,3,8), '.') \}

END_TEXT

Context("Vector");
WEIGHTED_ANS(basis_cmp([[$a[3],0,-$a[1]],[0,$a[3],-$a[2]]]),7);
WEIGHTED_ANS(basis_cmp([[$a[1],$a[2],$a[3]]]),3);
WEIGHTED_ANS(num_cmp($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);
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($T->element(3,3)),4);
WEIGHTED_ANS(num_cmp($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,);
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);

ENDDOCUMENT();       # This should be the last executable line in the problem.