Is "linear function" in linear algebra different from "linear function" in calculus? If so, why not use different words?

Question

Taking a single-variable function as an example, it seems to me linear function in linear algebra (also called linear transformation) is a function $f(x)$ that satisfies $f(x + y) = f(x) + f(y)$ and also $f(ax) = a f(x)$. But in Calculus a function will be called linear if it is of the form $f(x) = ax + b$. If this straight line passes through the origin, then it actually matches the linear algebra sense of the term.

Answer 1

Yes, they are different. In calculus (and high school mathematics before that), a linear function is defined as a map which takes $x$ to $mx+c$ for constants $m,c$. In linear algebra, we further restrict $c=0$. More precisely and generally, we say a function $f$ is linear if $f(x+y)=f(x)+f(y)$ for all $x,y$ in the vector space we are working in ($\mathbb R^n$ for example), as well as $f(cx)=cf(x)$ for all $x$ in the vector space we are in, and $c$ in the underlying field (think $\mathbb R^n$ with underlying field $\mathbb R$) we are in. You can tell that this definition rules out $x\mapsto mx+c$ with $c\neq0$. In the context of linear algebra, if we wanted to allow the constant to be nonzero, we call the mapping affine.