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

Thu Aug 16 23:07:34 2007 UTC (5 years, 9 months ago) by jj
File size: 1479 byte(s)
Moved problems back into place in the Rochester library, and set up
redirect files from their former locations.


    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
21 "PG.pl",
22 "PGbasicmacros.pl",
23 "PGchoicemacros.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