View of /trunk/NationalProblemLibrary/Rochester/setLinearAlgebra21InnerProductSpaces/ur_la_21_7.pg

    1 ##DESCRIPTION
    2 ##KEYWORDS('linear algebra', 'inner product', 'orthogonal', 'projection', 'function space')
    3 ##   hcao tagged and PAID on 2-20-2004
    4 ## DBsubject('Linear Algebra')
    5 ## DBchapter('Vector Spaces')
    6 ## DBsection('Inner Product')
    7 ## Date('6/3/2002')
    8 ## Author('')
    9 ## Institution('')
   10 ## TitleText1('Elementary Linear Algebra')
   11 ## EditionText1('5')
   12 ## AuthorText1('Larson, Edwards, Falvo')
   13 ## Section1('5.2')
   14 ## Problem1('')
   15 ##
   16 ##ENDDESCRIPTION
   17 
   18 DOCUMENT();        # This should be the first executable line in the problem.
   19 
   20 loadMacros(
   21 "PG.pl",
   22 "PGbasicmacros.pl",
   23 "PGchoicemacros.pl",
   24 "PGanswermacros.pl",
   25 "PGgraphmacros.pl",
   26 "PGnumericalmacros.pl",
   27 "PGstatisticsmacros.pl",
   28 "PGauxiliaryFunctions.pl"
   29 );
   30 
   31 TEXT(beginproblem());
   32 $showPartialCorrectAnswers = 1;
   33 
   34 $a = random(2,6,1);
   35 $b = non_zero_random(-5,5,1);
   36 
   37 ## 1 and sqrt(3)(1-2x) is an orthonormal basis for the subspace. Let i(x)=sqrt(3)(1-2x).
   38 ## prod_fh = $a/3 + $b;
   39 ## prod_fi = - $a*sqrt(3)/6;
   40 
   41 $proj = "- $a/6 + $b + $a x";
   42 
   43 BEGIN_TEXT
   44 
   45 Use the inner product \( \displaystyle <f,g>= \int_0^1 f(x)g(x)dx \) in the vector space \( C^0[0,1] \) to find
   46 the orthogonal projection of \(f(x) = $a x^2 + $b \) onto the subspace \( V \) spanned by \( g(x) = x \) and
   47 \( h(x) = 1 \).
   48 $BR
   49 (Caution: \( x\) and \(1\) do not form an orthogonal basis of \( V \).)
   50 $BR
   51 \( {\rm proj}_V(f) = \) \{ans_rule(20)\}.
   52 
   53 END_TEXT
   54 
   55 ANS(fun_cmp($proj));
   56 
   57 ENDDOCUMENT();       # This should be the last executable line in the problem.
   58