Which context of the two contexts of the linear function concept should be taught first?

In mathematics, the term linear function refers to two distinct but related notions:1

In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used.

In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.

Which context ("notion") of the two contexts of the linear function concept should be taught first?
Which one typically contain more basic math?
The Calculus context or the linear algebra context?

• Please clarify the level. Your question is only tagged linear algebra, but there are also tags for secondary education and undergraduate education.
– J W
Oct 23 '21 at 11:49

The first is taught first.

(In the US, typically) lines are taught in 9th grade Algebra 1. You learn to graph and write equations for lines. It's a huge part of the year.

The more advanced (but different) concept is from college/grad school classes. Linear algebra is a college class, sophomore year. Real analysis is taught junior year of college. Functional analysis is a graduate course.

I would add also that the term "linear" is used in differential equations for specific sorts of diffyQs. (Which are typically a required sophomore level college course, ODEs, and an optional junior level course, PDEs.)

I agree with Matt. The concepts are not confused by students. For one thing the kids who are in 9th grade, don't know all that other stuff coming later (which many will never have). And then the kids who do get these harder classes, later, with the use of the term linear, will have plenty of time/explanation to realize these are not y=mx+b beasts they are dealing with.

This isn't the only case where words may mean different things in different courses. Makes me think of this review: https://www.econjobrumors.com/topic/review-of-abstract-algebra-dummit-foote

The concepts are unrelated (at least in the context of mathematical teaching), and students are at least roughly familiar with both before they get to calculus or linear algebra.

• In calculus, a linear function from $$\mathbb R$$ to itself is of the form $$f(x)=ax+b$$, because it has a constant derivative and its graph is a straight line in the plane.
• In linear algebra, the only linear automorphism from $$\mathbb R$$ to itself is of the form $$f(x)=ax$$, because only that carries the linear structure of the domain into the range of the function.

In practice (at least in any linear algebra courses and textbooks I've read), there is no confusion between the two concepts. Those treatments refer to linear transformations or linear operators (or linear forms in more specific cases where the codomain is $$\mathbb R$$) instead of linear maps, so it's never struck me until just now that anyone would call it a linear function.