⁴I hope this is obvious visually or intuitively - if you move a sum of functions in space, then the result should be the same as if you moved the individual functions then added them together. This generalizes to any transformation of space. Since I want this document to be rigorous, a formal proof (apologies, it’s quite contrived): Consider the space of functions F, where each element f in F is of the form f : A → V . X can be any space, but V is a vector space (over a field K), which means F is also a vector space, with f₁ + f₂ defined such that (f₁ + f₂)(a) = f₁(a) + f₂(a) ∀a ∈ A. Suppose there is some transformation of the input to f T : A → A. We want to describe transforming f by way of transforming the space it acts on by T. To do so, let us define R : F → F, where R(f) := f ∘ T; in other words, R(f)(a) := f(T(a))  ∀a ∈ A. The claim is that R is linear on F. Proof: Consider f = xf₁ + yf₂ for x, y scalars in K. To prove R is linear, I claim R(f) = xR(f₁) + yR(f₂). By expansion: R(f) = R(xf₁ + yf₂). By definition of R, this equals (xf₁ + yf₂) ∘ T, which equals x(f₁ ∘ T) + y(f₂ ∘ T) by the linearity of functions - that is, (xf₁ + yf₂)(T(a)) = xf₁(T(a)) + yf₂(T(a))  ∀a ∈ A. Then, applying the definition of R, f₁(T(a)) = R(f₁) and f₂(T(a)) = R(f₂), so R(f) = xR(f₁) + yR(f₂)□. R here is also sometimes known as the pullback operator over T, such that R(f) is the pullback of f by T. There’s a connection to the transpose, but I don’t really understand it and this footnote is already wayyyyy too long.