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Find the orthogonal projection of the function $f(x)=x$ onto the subspace of $C([-1,1])$ spanned by $\sin\pi$x , $\sin2\pi$x, $\sin3\pi$x .

I was given this problem and I don't really know how to go about solving it. I understand orthogonal projection with vectors and can use Gram-Schmidt but I'm a bit confused how to even begin with this sort of problem. Any help would be much appreciated.

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Note that the inner product in $V=C([-1,1])$ is defined as $$ \langle f,g\rangle=\int_{-1}^1f(x)g(x)\ dx\tag{1} $$ Now you have three vectors in $V$, $$ v_1=\sin(\pi x), v_2=\sin(2\pi x), v_3=\sin(3\pi x). $$ And of course $f(x)=x$ is another vector in $V$. Denote it as $v=f$. Do you feel more familiar with this setting now and know how to go on?


Moreover, you don't need Gram-Schmidt here since $\{v_1,v_2,v_3\}$ is already an orthonormal set.


Exercise:

Calculate $\langle v,v_i\rangle$ for $i=1,2,3$ using (1). Then the projection is given by $$ a_1v_1+a_2v_2+a_3v_3 $$ where $a_i=\langle v,v_i\rangle$.

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  • $\begingroup$ OK I understand how to get the inner product of 2 functions but then is the projection of v onto $\left\{v_1, v_2, v_3\right\}$ just the vector with the 3 inner products as its entries? $\endgroup$ – B.K97 Jan 4 '17 at 22:48
  • $\begingroup$ @B.K97: Please see the edited answer. $\endgroup$ – Jack Jan 4 '17 at 22:52
  • $\begingroup$ OK I think I have it now thanks a lot $\endgroup$ – B.K97 Jan 4 '17 at 23:11
  • $\begingroup$ I was just doing a similar question and was wondering if the subspace {$v_1$,$v_2$,$v_3$} is not orthonormal would you then need to use Gram-Schmidt to find an orthonormal basis for the subspace in order to find the orthogonal projection? $\endgroup$ – B.K97 Jan 12 '17 at 20:35
  • $\begingroup$ @B.K97 Would you like to post a follow-up question? $\endgroup$ – Jack Jan 12 '17 at 22:36

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