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Why is the set of cosines and sines a vector space?

  1. Aug 18, 2015 #1
    So when I took linear algebra I was asked to show the well-known properties of orthogonality in the functions cos(mx) and sin(mx)... With this I mean all the usual combinations which I don't see why I should point out explicitly.

    The inner product in the vector space was defined ##\int_{-\pi}^{\pi} \phi_n \phi_m dx ## as usual.

    What I don't understand is why the space of functions ( cos(mx), sin(nx) ) (with this notation I'm pointing out all cosines and sines of integer angular frequency), is a vector space.

    Namely, its not closed under addition! If I add whichever combination I won't get a function of the form cos(mx) or sin(mx) back. There'll always be a phase constant ##\Phi## or something along those lines.

    Why are people so wishywashy with this? How can you define an inner product on a space of functions that doesnt form a vector space?

  2. jcsd
  3. Aug 18, 2015 #2
    The set of just those trig functions is not a vector space. However, the space of their linear combinations is. It is on this space (or perhaps a larger one like square integrable functions) that the inner product you described is defined. Here that exercise shows that [itex] 1,\sin x,\cos x, \sin 2x,\cos 2x,...[/itex] are orthogonal.
  4. Aug 18, 2015 #3
    Oh, that makes sense. I have one question though; when you have a set of functions like this one (infinite, orthogonal, not necessarily trigonometric) do they serve as a basis for all possible functions or not necessarily?

    If not, then is the Fourier Series so useful because the basis (1 sinx, cosx,.....cosnx,sinnx) describes a vector space that is "very big" ?
  5. Aug 18, 2015 #4
    Strictly speaking [itex] \{1,\sin x, \cos x,...\}[/itex] is only a basis for the space of their linear combinations. However, more functions are included in your space if you allow infinite sums of these trig functions. Not every function on [itex][-\pi,\pi][/itex]can be written as a (even infinite) sum of trig functions and the question of exactly which functions have a convergent Fourier series is quite complicated.
    Fortunately, 'nice' functions ([itex] C^1[/itex] is sufficient but not necessary) can be written as their Fourier series which is enough in many applications, the first historically being to the heat equation.
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