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The "Definition of a Linear Operator" is at the top of page 2 of the linked text.
My definition was missing that in order to be linear, A(cx) = cA(x). I mistakenly thought that this property was provable from the property I gave. Apparently it isn't because of "Hamel bases and the axiom of choice" (ChatGPT tried explaining.)
"straight-line property process" is not a helpful description of linearity for beginners or for professionals. "Linearity" is exactly when A(cx) = cA(x) and A(x+y) = A(x) + A(y). Describing that in words would be cumbersome. Defining it every time you see it is also cumbersome. When people come across "legitimate jargon", what they do (and need to do) is to learn a term when they need it to understand what they are reading and look up the definition if they forget.
I fully support experimental schemes to remove "illegitimate jargon" like medical latin, biology latin, and politic speak. Other jargon, like that in math and chemistry are necessary for communication.