Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, operator, or whatsoever) is. These different definitions can lead to confusion, especially when people apply conflicting definitions of linearity to the same mathematical object (just search on this website to figure out what I mean).
One of my favorite examples with potential for conflict is the natural logarithm $\ln : \mathbb{R}_{>0} \to \mathbb{R}$, which the majority of people, as I could learn during hot disputations not only here but also with colleagues in my field, regard as clearly non-linear, even though it fully complies with the definition of a linear map from the vector space $\mathbb{R}_{>0}$ to the vector space $\mathbb{R}$ (cf. Definition 4 below). A similar conflict arises analogously for the exponential function $\exp :\mathbb{R} \to \mathbb{R}_{>0}$.
I do not want to initiate yet another discussion here, about what the right and what the wrong definition of linearity is! Rather, ...
... I would like to compile a list of existing definitions, to get a better overview of the different uses of the term "linearity" across the different branches of mathematics and levels of education.
Let be begin with the following four, which are certainly far from constituting a complete list. What about bilinear or multilinear forms, linear morphisms and linear functors in category theory, linear metrics, or other concepts of linearity? I am looking forward to see your answers.
At school I was taught that
Definition 1: A function $f$ is linear if it is of the form $f(x) = ax + b$.
Later I learned that $f(x) = ax + b$ is affine, rather than linear, because
Definition 2: A function $f$ is linear if it is of the form $f(x) = ax$.
From real analysis and systems theory, I know the following
Definition 3: A map $f$ is linear if
- $f(x + y) = f(x) + f(y)$
- $f(a \cdot x) = a \cdot f(x)$.
This definition also applies to the expected value in statistics, as
- $\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)$
- $\mathrm{E}(a \cdot X) = a \cdot \mathrm{E}(X)$.
In a linear algebra course, I learned for the first time that we must not forget to take into account that the vector-space operations of the domain and the codomain of a map $f$ may be dissimilar. In fact, if $(V, \oplus, \odot)$ and $(W, \boxplus, \boxdot)$ are two $\mathbb{K}$-vector spaces, then
Definition 4: A map $f : V \to W$ is linear if
- $f(x \oplus y) = f(x) \boxplus f(y)$
- $f(a \odot x) = a \boxdot f(x)$.
Notice that I deliberately used distinct symbols for the operations in both vector spaces, to emphasize how this definition differs from Definition 3.
Correct me if I am wrong, but this definition should also apply to maps between two modules.
Updates: