Apart from specific mathematical uses, "linear" (from Latin linea "line") means "in the form of/having to do with a straight line". A linear equation is obviously called that because its graph is a straight line.

But why is a linear map, which is a different thing, called "linear"? What is it about a linear map that is "having to do with a straight line"?

Please note: I know that "linear function", and the word "linear" in general, can mean these two different things:

  • A function, such as $f(x)=ax+b$, that has a straight-line graph on the Cartesian plane, also called a "linear equation"
  • A function obeying the constraints $f(\mathbf{a}+\mathbf{b})=f(\mathbf{a})+f(\mathbf{b})$ and $f(k\mathbf{a})=kf(\mathbf{a})$, also called a "linear map"

I am not asking for a definition of either of these things. Nor am I asking whether or not they're the same. I know they aren't.

I am trying to be clear about this because there seems to be confusion. There is a very similarly-worded question title already, "Why is a linear transformation called linear?" That's marked as a duplicate of another question, "Why are linear functions linear?". And my question was closed as a duplicate too, but I'm positive that it's not. You see, both of those are actually about confusion between those two things, and the answers are "they're different, don't get them confused, they just use an ambiguous name".

My question is, why is the name "linear" applied to the second concept at all? What is "linear" about it? Why wasn't it called a "proportional", or "fandangled", or "dinglehopper" map instead? I want to know why it is that "linear" is an applicable term for this thing we call a "linear map".

Reasons that don't seem to suffice, but please correct me if I'm wrong on any of these:

  • Because its graph is a straight line. (Not good enough, because it's specifically a straight line through the origin, not a general straight line. Straight lines in general are affine maps, AKA linear equations in the other sense.)
  • Because it preserves straight lines. (Unless I'm mistaken, so does an affine map.)
  • Because it bears the property that we call "linearity". (Circular reasoning. Which came first, linearity or the linear map? Whichever it is, I want to know why that was called "linear".)
  • $\begingroup$ A linear map sends straight lines to straight lines. $\endgroup$ – Morgan Rodgers Nov 26 '18 at 12:25
  • $\begingroup$ @MorganRodgers: Correct me if I'm wrong, but doesn't an affine map also send straight lines to straight lines? It doesn't preserve parallel lines, but they stay straight... don't they? $\endgroup$ – Tim Pederick Nov 26 '18 at 12:29
  • $\begingroup$ but if you primarily care about lines through the origin.... $\endgroup$ – Morgan Rodgers Nov 26 '18 at 13:14
  • $\begingroup$ Maybe asking at hsm.stackexchange.com will generate some answers with actual evidence in them (rather than mere opinions). $\endgroup$ – GEdgar Nov 27 '18 at 13:11
  • $\begingroup$ @GEdgar: Thanks, I didn't even know that particular SE existed! I was hoping there was some obvious (or at least well-understood) reason that I was missing, but it seems not, so asking about the history will be worth a shot. $\endgroup$ – Tim Pederick Nov 28 '18 at 9:08

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