Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read:
"A linear function is a function defined by an equation of the form $$y = Ax + B."$$
This strikes me as a common definition, and while this is certainly an alright algebraic definition for the equation of a line, it is certainly not consistent with the definition of a linear function in general, which I give as
"A linear function $f$ is a rule of correspondence from a set $A$ to a set $B$ such that
1.$f$ relates to each element of $A$ a unique element of $B$,
2. $f(a+b) = f(a) + f(b)$
3. $f(a \cdot x) = a \cdot f(x)$"
We take for granted that $+$ and $\cdot$ are carried out with respect to their respective sets and need not coincide.
In this more general definition the elements need not be real numbers. Indeed, one can show some linear functions that take as their arguments complex numbers, matrices, vectors, other functions, and even more exotic objects.
It's easy to show that a line with equation $y = Ax$ satisfies both definitions, but $y = Ax +B$ fails to satisfy the general definition.
Today, I decided to tell my students that "every line is a translation of the graph of a linear function over the real numbers."