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Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read:

"A linear function is a function defined by an equation of the form $$y = Ax + B."$$

This strikes me as a common definition, and while this is certainly an alright algebraic definition for the equation of a line, it is certainly not consistent with the definition of a linear function in general, which I give as

"A linear function $f$ is a rule of correspondence from a set $A$ to a set $B$ such that

1.$f$ relates to each element of $A$ a unique element of $B$,
2. $f(a+b) = f(a) + f(b)$
3. $f(a \cdot x) = a \cdot f(x)$"

We take for granted that $+$ and $\cdot$ are carried out with respect to their respective sets and need not coincide.

In this more general definition the elements need not be real numbers. Indeed, one can show some linear functions that take as their arguments complex numbers, matrices, vectors, other functions, and even more exotic objects.

It's easy to show that a line with equation $y = Ax$ satisfies both definitions, but $y = Ax +B$ fails to satisfy the general definition.

Today, I decided to tell my students that "every line is a translation of the graph of a linear function over the real numbers."

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    $\begingroup$ Unless you have an unusually high percentage of students who will reach the level of elementary linear algebra ("unusually high percentage" for U.S. classes is probably anything over 15%), you're better off avoiding any mention of "linear" in the linear algebra sense. Also, since the origin is usually not all that physically significant in applications, lines are sufficiently linear for your audience, since (when not horizontal or vertical) they give rise to functions in which output changes are directly proportional to input changes. $\endgroup$
    – Dave L Renfro
    Oct 26, 2015 at 20:29
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    $\begingroup$ A common term for such line-like functions other than "linear" would be "affine". $\endgroup$
    – quid ♦
    Oct 26, 2015 at 22:52
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    $\begingroup$ Personally I would avoid using the idea of a 'linear function' for pre-calc at all. We always just talked about lines, or 'the equation of a line'. $\endgroup$
    – Jessica B
    Oct 27, 2015 at 6:33
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    $\begingroup$ Given what you said about the students you're teaching, my concerns drop quite a bit. Incidentally, something to be aware of is that "quadratic function" is used for degree two polynomials, but "linear function" (in the precalculus sense) usually DOES NOT mean degree one polynomials, since a constant function is not a degree one polynomial. See this 27 September 2007 post for some comments about lines that may be of interest. $\endgroup$
    – Dave L Renfro
    Oct 28, 2015 at 14:08
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    $\begingroup$ In France, we use "linaire" for $f(x) = ax$ and "affine" for $f(x) = ax +b$. $\endgroup$
    – projetmbc
    Nov 10, 2015 at 10:29

7 Answers 7

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The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, restricted notion.

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    $\begingroup$ "more restricted"? The modern notion is, if anything, more general. $\endgroup$
    – Tim Seguine
    Oct 28, 2015 at 15:06
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    $\begingroup$ It's more restricted in the sense that some functions (like f(x) = mx + b) that would have been considered linear under the "older" sense of the word are not linear in the modern sense. It's more general in the sense that it allows you to talk about linear mappings on arbitrary vector spaces, not just functions $\mathbb{R} \to \mathbb{R}$. $\endgroup$
    – mweiss
    Oct 28, 2015 at 15:12
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You can disambiguate the two senses of ‘linear’ as ‘homogeneous linear’ and ‘affine linear’. You can then say that in this class, ‘linear’ means ‘affine linear’ by default but in other situations it will mean ‘homogeneous linear’ instead.

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The function $y(x)=ax+b$ is linear under homogeneous coordinates; more generally, $y = g(x_1,x_2,...,x_k)=b_0+b_1x_1+b_2x_2+...+b_kx_k$ is linear in that sense.

Specifically, rendering $g$ in homogeneous coordinates,

$g(x_1,x_2,...,x_k)=f(x_1,x_2,...,x_k,1)$,

and $y=f(x_1,x_2,...,x_k,1)$ is linear in the usual sense.

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    $\begingroup$ Good answer. An advantage of this perspective is that it also captures vertical lines $x=c$. $\endgroup$
    – user52817
    Oct 27, 2015 at 20:21
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    $\begingroup$ Can you add a bit more elucidation to this answer? $\endgroup$
    – Andrew
    Nov 6, 2015 at 13:55
  • $\begingroup$ Possibly -- what were you wanting to see elucidated in particular? $\endgroup$
    – Glen_b
    Nov 6, 2015 at 15:57
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    $\begingroup$ Just as $y=ax$ is homogeneous-linear and the more general $y=ax+b$ is affine-linear, so the yet more general $Ax+By=c$ (which, as @user52817 noted, can also be captured by a homogenous-linear $f$) may be called projective-linear. $\endgroup$
    – Toby Bartels
    Mar 21, 2021 at 15:30
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I believe the distinction that's bothering you is now called "direct variation." This term is used to describe a line that goes through the origin, i.e. Y=mX, and is considered a subset of linear functions Y=mX+b. I graduated HS in 1980, and don't recall direct variation being mentioned. Now, our local HS math curriculum includes it, but the HS a few towns away where I tutor doesn't seem to mention it.

You mention precalc which can either be in high school or college. In my opinion, math students struggle enough with the material, and adding a layer of these distinctions might not be appropriate at this stage.

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In math and physics people tend to name things similarly in different field's and sub-fields.

It would be best to stop using the term linear function and instead call them linear equations, for polynomials of order zero or one, and linear operators [or some other term depending on what branch of science you come from], for operators where f(a+b) = f(a) + f(b).

Also, I would explain to your students that many terms have multiple definitions (e.g. Euler's numbers) or multiple terms can refer to a single definition (Normal/Gaussian distribution). They will see this often through out their academic career. It only gets worse as you delve deeper into scholarly pursuits.

The number of things named after Euler is a good example of this pattern, where in, there are 4+ Euler numbers, a multitude of theorems, and various synonyms (Euler's formula =/= Euler's equation).

https://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler#Euler.27s_formulas

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    $\begingroup$ Thanks Greg. I've been stressing the difference between a linear function and a linear equation since I asked the question. In our math club meeting this week, we're going to discuss linear and affine transformations, and I'm going to show that all of our standard geometric shapes (circle, ellipse, square, rhombus, isosceles triangle, right triangle, etc) are affine transformations of either a circle or a regular polygon. $\endgroup$
    – Andrew
    Nov 3, 2015 at 20:25
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    $\begingroup$ I don't think this terminology is a good idea: it uses words corresponding to objects of different nature (equation vs operator) to distinguish between two different properties, each of which makes sense for each object. If you let me use the adjectives linear and affine (for what have been also called homogeneous linear and affine linear by Toby Bartels), then there are such things as linear operators, affine operators, linear equation, affine equations. Whatever terminology is chosen it should not get in the way of distinguishing between an operator and an equation. $\endgroup$
    – Benoît Kloeckner
    Apr 26, 2017 at 13:59
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It isn't just in elementary mathematics that functions of the form $f(x) = mx+b$ are called linear. The term "linear regression" is used almost universally in statistics to describe the method of fitting a line to a data set. I don't remember ever once seeing the term "affine regression" used instead.

Even though mathematics can be modeled as a formal language/system such as ZFC, it is actually done in a formal language/ natural language hybrid, one which is not immune from the sort of etymological quirks that characterize all natural languages.

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Electronic engineers describe a linear amplifier or other linear device as one obeying Output = Input x Gain (or loss), so if input = 0 then output = 0, as in "direct variation" above used to describe a line that goes through the origin, i.e. Y=mX.

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    $\begingroup$ This answer does not answer the question. $\endgroup$
    – Tommi
    Aug 31, 2017 at 5:39
  • $\begingroup$ I was merely giving my pennyworth from an engineering pov. I claim minimal competence in mathematics. $\endgroup$
    – user476046
    Apr 23, 2018 at 10:14
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    $\begingroup$ This website is for asking specific questions and for answering them, rather than for general forum-type discussions and comments. If you can expand what you have into an answer to the question asked here, it is more than welcome. $\endgroup$
    – Tommi
    Apr 23, 2018 at 14:47

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