DOCUMENT();

loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "scaffold.pl",
  "parserMultiAnswer.pl",
  "rank.pl",
  "CustomBasisChecker_Rows.pl",
  "weightedGrader.pl",
  "PGcourse.pl"
);

TEXT(beginproblem());

install_weighted_grader();

$showPartialCorrectAnswers = 1;
$showPartialCredit = 1;

# Set the Numeric context, but forbid using the variable x in answers.
Context("Numeric");
Context()->parens->set("[" => {formMatrix => 1});

Context()->variables->remove("x");

Context()->flags->set(
  tolerance => 0.00000001,
  tolType => "absolute",
);

# ====================================================


# Space where I had problems had correct basis as (1,0,3,4) and (0,1,-1,-1)
# The answer (1,2,1,2) and (sqrt(253),2sqrt(253),sqrt(253),2sqrt(253)+t).

@given1 = ( [ 1, 0,  3, 4 ] );
@given2 = ( [ 1, 2,  1, 2 ] );
@given3 = ( [ 1, 3,  0, 1 ] );
@given4 = ( [ 1, 4, -1, 0 ] );

$g1 = Matrix( @given1 );
$g2 = Matrix( @given2 );
$g3 = Matrix( @given3 );
$g4 = Matrix( @given4 );

# Sample basis = 2 independent vectors in the space (has dim=2)

$vec1 = Matrix([[ 1, 0,  3,  4 ]]);
$vec2 = Matrix([[ 0, 1, -1, -1 ]]);

###########################################
#  The scaffold
Scaffold::Begin(
      can_open => "when_previous_correct",
      is_open  => "correct_or_first_incorrect",
      instructor_can_open => "always",
      after_AnswerDate_can_open => "always",
      hardcopy_is_open  => "correct_or_first_incorrect",
      allow_next_section_to_open_on_preview => 0,
      prevent_close_by_preview => 1,
    );


###########################################
Section::Begin("Part 1: Calculate the dimension");

$ans1 = Real("2");

Context()->texStrings;
BEGIN_TEXT

Consider the following vectors which belong to the vector space
\( V = \mathbb{R}^4 \):

\[ \begin{array}{lcl}
     v_1 & = & $g1 \\
     v_2 & = & $g2 \\
     v_3 & = & $g3 \\
     v_4 & = & $g4 
   \end{array}
\]

Let \( U \) be the subspace of \( V \) spanned by these vectors:
$BCENTER
\( U = \text{span} \lbrace v_1, v_2, v_3, v_4 \rbrace \).
$ECENTER

$PAR

The dimension of \( U \) is:
\( \text{dim} \; U = {} \)
\{ $ans1->ans_array(3) \}.
$PAR

END_TEXT
Context()->normalStrings;

WEIGHTED_ANS( $ans1->cmp(), 10 );

Section::End();

###########################################
Section::Begin("Part 2: Find a basis");

Context('Matrix');

$multians1 = MultiAnswer($vec1, $vec2)->with(
  singleResult => 1,
  separator => ',',
  tex_separator => ',',
  allowBlankAnswers=>0,
  checker => ~~&basis_checker_rows_tani,
);

Context()->texStrings;
BEGIN_TEXT
Enter a basis \( \mathcal{B} \) of \( U \) below.
$BR
$BCENTER
\( \mathcal{B} = \left\lbrace \;\; \rule{0pt}{20pt} \right. \)
\( u_1 = \)
\{ $multians1->ans_array(3) \} \( , \)
$BR
\( \rule{60pt}{0pt} u_2 = \)
\{ $multians1->ans_array(3) \}
\( \left. \;\; \rule{0pt}{20pt} \;\; \right\rbrace \)
$ECENTER

END_TEXT
Context()->normalStrings;

WEIGHTED_ANS( $multians1->cmp(), 90 );

Section::End();

Scaffold::End();

ENDDOCUMENT();