<problem>
<script type="loncapa/perl">
$a1 = random(-6,6,4);
$a2 = random(-6,6,4);
$n1 = random(3,11,2);
$n2 = random(2,10,2);
$function = "$a1*cos($n1*x)+$a2*sin($n2*x)";
$example=&xmlparse('An example would be <m eval="on">$(sin($n1\cdot x)+cos($n2\cdot x))/\sqrt{2}$</m>');
</script>
<startouttext />
Give an example of a function
<ol>
<li>which is orthogonal to <algebra>$function</algebra> with respect to the
scalar product
<m>\[<g \mid h> = \frac{1}{\pi} \int_{-\pi}^{\pi}dx g(x) \cdot h(x)\]</m>
</li>
<li>whose norm is 1.</li>
</ol>
<endouttext />
<mathresponse answerdisplay="$example" cas="maxima" args="$function">
<answer>
overlap:integrate((RESPONSE[1])*(LONCAPALIST[1]),x,-%pi,%pi)/%pi;
norm:integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(overlap=0 and norm=1);
</answer>
<textline readonly="no" size="50" />
<hintgroup showoncorrect="no">
<mathhint name="ortho" args="$function" cas="maxima">
<answer>
overlap: integrate((LONCAPALIST[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(not overlap = 0);
</answer>
</mathhint>
<mathhint name="norm" args="$function" cas="maxima">
<answer>
norm: integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(not norm = 1);
</answer>
</mathhint>
<hintpart on="norm">
<startouttext />
The function you have provided does not have a norm of one.
<endouttext />
</hintpart>
<hintpart on="ortho">
<startouttext />
The function you have provided is not orthogonal.
<endouttext />
</hintpart>
</hintgroup>
</mathresponse>
<postanswerdate>
<startouttext />
<p>
Note that with respect to the above norm, <m>$\cos(nx)$</m> is
perpendicular to <m>$\sin(nx)$</m> and perpendicular to <m>$\cos(mx)$</m>
for <m>$n\ne m$</m>.
</p>
<endouttext />
</postanswerdate>
</problem>
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