How is the Fourier transform "linear"?

A "linear" function usually means one who's graph is a straight line, or that involves no powers higher than 1. And yet, many sources will tell you that the Fourier transform is a "linear transform".

Both the discrete and continuous Fourier transforms fundamentally involve the sine and cosine functions. These functions are about as non-linear as you can get. They are transcendental. They are solutions to differential equations. They are infinite-degree polynomials. They have infinity turning points. (So, not straight lines at all.) In the discrete case, you take a finite sum with them, but in the continuous case you compute an integral with them! This all sounds highly non-linear to me...

Then again, apparently "linear" is one of those mathematical terms that has many, many meanings. (Much like "normal" means a few trillion different things depending on the exact context in question.) And perhaps not all of those meanings are actually related.

So, my question is, in which exact sense is the Fourier transform considered "linear"?

Let $f$, $g$ be functions of a real variable and let $F(f)$ and $F(g)$ be their fourier transforms. Then the fourier transform is linear in the sense that, for complex numbers $a$ and $b$,

$$F(af + bg) = a F(f) + b F(g)$$

i.e. it has the same notion of linearity that you may be used to from linear algebra. This is not a quirk - it expresses the fact that functions form an infinite dimensional vector space, with addition and multiplication by a scalar defined in the obvious way:

$$(f+g)(x) = f(x) + g(x)$$ $$(af)(x) = a f(x)$$

Linear in this context has nothing to do with the geometry of the graphs of functions under consideration. It is used in the sense that the functions themselves form a vector space, i.e., you can add functions and take multiples of a function with a real number. The linearity of the fourier transform means that if you take the transform of a sum of functions, it is the same as the sum of the fourier transforms of the functions, and the same holds for real multiples of functions.