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".)
