Linear functions/operators conflicting understanding

[Jonnyb42]

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

 

[bp_psy]

It is linear if b=0 otherwise is an affine transformation.
http://en.wikipedia.org/wiki/Affine_map