Always blew my mind that every signal can be recreated simply by adding different sine waves together.
replies(5):
Turns out... they are not! You can do the same thing using a different set of functions, like Legendre polynomials, or wavelets.
Which to your point: You're absolutely correct that you can use a bunch of different sets of functions for your decomposition. Linear algebra just says that you might as well use the most convenient one!
For someone reading this with only a calculus background, an example of this is that you get back a sine (times a constant) if you differentiate it twice, i.e. d^2/dt^2 sin(nt) = -n^2 sin(nt). Put technically, sines/cosines are eigenfunctions of the second derivative operator. This turns out to be really convenient for a lot of physical problems (e.g. wave/diffusion equations).