** Using WeBWorK **

## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## DBsubject(Linear algebra)
## DBchapter(Matrix factorizations)
## DBsection(Jordan form)
## Date(11/17/2016)
## Institution(NAU)
## Author(Nandor Sieben)
## Level(4)
## MO(1)

DOCUMENT();
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "parserMultiAnswer.pl",
  "AnswerFormatHelp.pl",
  "PGchoicemacros.pl",
  "MatrixReduce.pl",
  "rank.pl",
  "PGcourse.pl"
);
$showPartialCorrectAnswers = 1;
TEXT(beginproblem()); 

Context("Numeric");

$a=1;
$b = $a+random(0,2,1);
$c = $b+random(0,2,1);
$d = $c+random(1,3,1);
$e = $d+random(0,1,1);
$f = $e+random(0,1,1);
$answer=List($a,$b,$c,$d,$e,$f);
@row1=("\( k \)");
@row2=("\( \text{nullity}(L_\lambda^k)  \) ");
$s[0]=0;
for ($i=1; $i < 20; $i++) {
  $s[$i]=$s[$i-1]+($a>=$i)+($b>=$i)+($c>=$i)+($d>=$i)+($e>=$i)+($f>=$i);
  if ($s[$i] > $s[$i-1] ) {
    push @row1, "$i";
    push @row2, "$s[$i]";
  }
}

$table_start = begintable(4);
$table_row1  = row( @row1 );
$table_row2  = row( @row2 );
$table_end   = endtable();

Context()->texStrings;
BEGIN_TEXT
Let \( \lambda \) be an eigenvalue of the linear operator \( L \) and define \( L_\lambda := L - \lambda I \). 
The following table lists the nullities of the powers of \( L_\lambda \).
$BCENTER
$table_start
$table_row1
$table_row2
$table_end
$ECENTER
Find the sizes of the Jordan blocks corresponding to \( \lambda \) of the Jordan form  of the matrix of \( L \) as a list of integers. 
$BR
Sizes: \{ ans_rule(20) \}


END_TEXT
Context()->normalStrings;

ANS( $answer->cmp() );


COMMENT('');

ENDDOCUMENT();