Definitions of "linearity" across branches of mathematics or levels of math education

Question

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:

Answer 1

Not exactly an solution to your question, but frankly it's too big to be fitting a comment. The distinction as to exactly what is being linear is very important. It is possible to approximate non-linear behaviours if we are allowed to make the linear space (as in linear algebra) large enough. Approximating solutions to smaller dimensional non-linear equations by large dimensional matrix equations or least-squares problems.

---

One of possibly simplest examples would be a binary decision. Something should "activate" once temperature goes above some level, and we have an in-signal (variable in-value) saying "increase", "decrease" by one "notch" or to "stay the same". We would like to activate for example a fan if things are getting hot enough. So our function value is 0 or 1. Such a "step" function (like the Heaviside step function) is highly non-linear, but we can still describe it with a (high dimensional) linear state space and realize it with, say a slightly modified cyclic permutation matrix $\bf P$ (it should not be allowed to "flip around" but rather be "saturated" at the max and min temp ends). Then counting up would be multiplication with ${\bf P}^1$, counting down ${\bf P}^{-1}$ and staying the same ${\bf P}^0$.

---

EDIT Oh bollocks, I forgot the punch-line. Our out function would then be scalar product with ${\bf v} = [0,\cdots, 0, 1, \cdots ,1]$ to select the appropriate function value. ${\bf v}^T{\bf s}$, where $\bf s$ contains the state and $\bf v$ the function values and the $\bf s$ is updated as ${\bf s} = {\bf P}^k{\bf s}$ for the in-value increase (k=1), decrease (k=-1) or stay the same (k=0).

Answer 2

I just recall a definition from calculus:

Definition: A function $f : I \subseteq \mathbb{R} \to \mathbb{R}$ that is differentiable on the interval $I$ is linear if $\dfrac{\mathrm{d}f}{\mathrm{d}x} = c$ for a constant $c \in \mathbb{R}$.