The Fourier Transform can also be thought of as part of Linear Algebra, because it's actually funding a representation of a given function in the basis consisting of sin and cos functions (or complex exponentials). See, the collection of non-pathological functions is a vector space. We add elements by adding the functions pointwise, we multiply by a constant in the obvious way, and the other requirements can be check…
You have indicated quite explicitly that you're being informal (by using non-technical terms like 'non-pathological functions'), but I think it's worth making the small observation here that sine and cosine functions do not form a basis (implicitly: "a Hamel basis") for, say, the space of continuous functions on R/Z in the usual sense of linear algebra, since not every function can be written as a finite linear combination of them.
Rather, they are a topological basis, in the sense that every function can be written as an infinite linear combination of them. Why is it worth making this point, which probably seemed too obvious to say? The reason is that, unlike finite sums, which are unambiguous algebraic constructs, infinite sums require topology to compute; and being a topological basis in, say, the L^2 topology is quite different to being a topological basis in, say, the topology of pointwise convergence. The study of the different kinds of summability of Fourier series is the subject of some very, very deep work.