The orthogonality is essentially follows from (1) integer frequency complex sinusoids have an average value of zero over [0,2π], and (2) if you multiply two distinct integer frequency complex sinusoids, you get another integer frequency complex sinusoid. I'm not sure that this is any more intuitive.
> what would be useful is to give an example of a set of functions which are not linearly independent.
See [1,2] for example, which (I believe) has applications in compressed sensing and dictionary learning.
>The orthogonality is essentially follows from (1) integer frequency complex sinusoids have an average value of zero over [0,2π], and (2) if you multiply two distinct integer frequency complex sinusoids, you get another integer frequency complex sinusoid. I'm not sure that this is any more intuitive.
i think these kinds of explanations are hilariously pointless. and i don't mean to disparage because you're just trying to answer op's question but all you've done is restated the proof in english - i.e. of course it follows from that because what you've just said is the inner product of basis functions is 0. well yes of course that's definition of orthogonal.
> what would be useful is to give an example of a set of functions which are not linearly independent.
See [1,2] for example, which (I believe) has applications in compressed sensing and dictionary learning.
[1]: https://en.wikipedia.org/wiki/Frame_(linear_algebra)
[2]: https://en.wikipedia.org/wiki/Overcompleteness