## DESCRIPTION
##   Fundamental Subspaces of a Matrix
## ENDDESCRIPTION

## KEYWORDS('Nullspace', 'Column space', 'Row space', 'Left nullspace')
## 

## DBsubject('Calculus')
## DBchapter('')
## DBsection('')
## Date('10/04/2009')
## Author('Ted Shifrin')
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')
           
DOCUMENT();	# This should be the first executable line in the problem.

loadMacros("PGstandard.pl",      
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",      
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "PGmatrixmacros.pl",
           "PGmorematrixmacros.pl",
           "compoundProblem.pl",
           "PGnumericalmacros.pl",
          );      
         
     
TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = non_zero_random(-3, 3, 1);
$b = non_zero_random(-3, 3, 1);
$c = non_zero_random(-3, 3, 1);
$d = non_zero_random(-3, 3, 1);
$d1 = non_zero_random(-3, 3, 1);
$e = non_zero_random(-3, 3, 1);
$f = non_zero_random(-3, 3, 1);
$i = non_zero_random(-1, 1, 2);
$g = $e*$f+$i;
$k = non_zero_random(-3, 3, 1);
$l = non_zero_random(-3, 3, 1);
$m = non_zero_random(-3, 3, 1);
$n = non_zero_random(-3, 3, 1);


$m11=1; $m12=$a; $m13=$e;  $m14=$b+$e*$d; $m15=$c+$e*$d1;
$m21=$f; $m22=$f*$a; $m23=$g; $m24=$f*$b+$g*$d; $m25=$f*$c+$g*$d1;
$m31=$k; $m32=$k*$a; $m33=$l; $m34=$k*$b+$l*$d; $m35=$k*$c+$l*$d1;
$m41=$m; $m42=$m*$a; $m43=$n; $m44=$m*$b+$n*$d; $m45=$m*$c+$n*$d1;

$LN11=-$k+$i*$f*($l-$e*$k);
$LN12=-$i*($l-$e*$k);
$LN13=1;
$LN14=0;

$LN21=-$m+$i*$f*($n-$e*$m);
$LN22=-$i*($n-$e*$m);
$LN23=0;
$LN24=1;

$isProfessor = ($studentLogin eq 'shifrin' || $studentLogin eq 'test');

$cp = new compoundProblem(
  parts=>2,
  weights=>[.4,.6],
  parserValues=>1,
  allowReset => $isProfessor,
  nextVisible => 'Always',
  nextStyle => 'Button',
);

$part = $cp->part;

if($part==1){
BEGIN_TEXT

Consider the matrix

\[ A = \left[\begin{array}{r r r r r} 
$m11 & $m12 & $m13 & $m14 & $m15\cr
$m21 & $m22 & $m23 & $m24 & $m25 \cr
$m31 & $m32 & $m33 & $m34 & $m35 \cr 
$m41 & $m42 & $m43 & $m44 & $m45 
\end{array} \right]\quad .
\]

$PAR
We start with some basic facts about the four fundamental subspaces
associated to \(A\):
$PAR
\(\mathbf R(A) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
$BR
\(\mathbf C(A) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
$BR
\(\mathbf N(A) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.
$BR
\(\mathbf N(A^T) \) is a subspace of \(\mathbb R^k\) with \( k = \) \{ans_rule(3)\}.

$PAR
Give the reduced echelon form of \(A\). 
$BR
\{ mbox( answer_matrix(4,5,5) )\}


$PAR
Give the dimension of each of the fundamental subspaces:
$BR
dim \(\mathbf R(A) = \) \{ ans_rule(5) \}
$BR
dim \(\mathbf C(A) = \) \{ ans_rule(5) \}
$BR
dim \(\mathbf N(A) = \) \{ ans_rule(5) \}
$BR
dim \(\mathbf N(A^T) = \) \{ ans_rule(5) \}
$PAR

When you have answered all these questions correctly, click on the "Go on to next part" button.

END_TEXT

ANS(num_cmp(5)); ANS(num_cmp(4)); ANS(num_cmp(5)); ANS(num_cmp(4));

ANS(num_cmp(1)); ANS(num_cmp($a)); ANS(num_cmp(0)); ANS(num_cmp($b)); ANS(num_cmp($c));
ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(1)); ANS(num_cmp($d));
ANS(num_cmp($d1));
ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0));
ANS(num_cmp(0));
ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0)); ANS(num_cmp(0));
ANS(num_cmp(0));

ANS(num_cmp(2)); ANS(num_cmp(2)); ANS(num_cmp(3)); ANS(num_cmp(2));

# $cp->useGrader(~~std_problem_grader);
}

if($part==2){
BEGIN_TEXT

Recall the original matrix and your reduced echelon form:

\[ \hbox{ \[ A = \left[\begin{array}{r r r r r} 
$m11 & $m12 & $m13 & $m14 & $m15\cr
$m21 & $m22 & $m23 & $m24 & $m25 \cr
$m31 & $m32 & $m33 & $m34 & $m35 \cr 
$m41 & $m42 & $m43 & $m44 & $m45 
\end{array} \right]}\quad \text{and} \quad \hbox{\left[\begin{array}{r r r r r} 
1 & $a & 0 & $b & $c\cr
0 & 0 & 1 & $d & $d1\cr
0 & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & 0
\end{array} \right]}\quad .
\]

$PAR
Give bases for each of the four fundamental subspaces.
$BR
\{mbox('\(\mathbf R(A)\):', ans_array(5,1,8),',', ans_array_extension(5,1,8) ) \}
$BR
\{mbox( '\(\mathbf C(A)\):', ans_array(4,1,8),',', ans_array_extension(4,1,8) ) \}
$BR
\{mbox( '\(\mathbf N(A)\):',  ans_array(5,1,8),',', ans_array_extension(5,1,8), ',', ans_array_extension(5,1,8) ) \}
$BR
\{mbox('\(\mathbf N(A^T)\):', ans_array(4,1,8),',', ans_array_extension(4,1,8) )\}
$BR

END_TEXT

ANS(basis_cmp([[1,$a,0,$b,$c],[0,0,1,$d,$d1]]));
ANS(basis_cmp([[$m11,$m21,$m31,$m41],[$m13,$m23,$m33,$m43]]));
ANS(basis_cmp([[-$a,1,0,0,0],[-$b,0,-$d,1,0],[-$c,0,-$d1,0,1]]));
ANS(basis_cmp([[$LN11,$LN12,$LN13,$LN14],[$LN21,$LN22,$LN23,$LN24]]));

}

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