Linear functions/operators conflicting understanding

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The discussion centers on the definition of linear functions and operators in theoretical physics, specifically addressing the conditions that classify a function as linear: f(ax) = a f(x) and f(x + y) = f(x) + f(y). A participant expresses confusion regarding the linear equation y = mx + b, which does not satisfy these conditions unless b equals zero, thus categorizing it as an affine transformation. The distinction between linear and affine transformations is clarified, emphasizing that linearity requires the absence of a constant term.

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This discussion is beneficial for students in theoretical physics, mathematicians, and anyone seeking to clarify the concepts of linear and affine functions in mathematical contexts.

Jonnyb42
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So in my theoretical physics class my professor was reminding us ("reminding") of what "linear" means, such as a linear functions or a linear operator. He said the definition was:

1. f(ax) = a f(x)

2. f(x + y) = f(x) + f(y)

Functions or operators are "linear" if they meet the above 2 conditions.
I am confused because, if I take a line... y = mx + b, which I completely consider to be linear, it doesnt' match either of those conditions.

Are there 2 different kinds of linear, am I losing my mind... or what?

thanks
 
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