I think he meant the 'decomposition' definition. You're decomposing your signal into other signals, that's the fundamental aspect. This decomposition is a linea

I think he meant the 'decomposition' definition. You're decomposing your signal into other signals, that's the fundamental aspect. This decomposition is a linear, so you can also make it so your new components have certain nice properties -- they can be orthogonal, unitary, etc. Then you naturally get waves (complex exponentials) as the components. And then you realize -- hey, those are exponentials -- so the rule that integrating/differentiating each component is trivial allows you to solve linear ODE really quickly. I don't think algebra gets in the way at all -- it just makes this decomposition crystal clear. It makes generalizing easy also -- there are other nice basis like cosine basis, and so on. You can choose any N functions to decompose your N-point function, as long as they are linearly independent, you're guaranteed perfect reconstruction.