TITLE: Why is a linear transformation called linear?
QUESTION [38 upvotes]: $T(av_1 + bv_2) = aT(v_1) + bT(v_2)$
Why is this called linear? $f(x) =ax + b$, the simplest linear equation does not satisfy $f(x_1 + x_2) = f(x_1) + f(x_2)$.
Thank you.

REPLY [3 votes]: You may refer to them as "$k$-vectorspace homomorphisms" if you wish.
In a little more detail: if $k$ is a field, then a $k$-vectorspace can be thought of as a set $X$ equipped with, for each sequence $a \in k^n$, a corresponding "linear combination mapping" $X \leftarrow X^n$. By convention, the structure-preserving mappings between $k$-vectorspaces are called "$k$-linear transforms," or simply $k$-linear, probably because the phrase "linear-combination-mapping-preserving mapping" is altogether too confusing. Like I said, you may refer to them as $k$-vectorspace homomorphisms if you wish.
But this just pushes the issue back: why are they called, of all things, linear combinations? Well, if $X$ is a $k$-vectorspace we're given a single vector $x \in X$, then $$\{ax : a \in k\}$$ is the set of all elements of $X$ that can be obtained by applying linear combination mappings to $x.$ In the special case where $k=\mathbb{R}$, this can be visualized as a line through the origin.
So "linear combination" is probably best thought of as a shorthand for "line-through-the-origin combination." Of course, if we've got two vectors, then we're potentially talking about a plane through the origin. And so on.
In fact, linear algebra isn't really about lines at all; its really about flat things that pass through the origin. Perhaps it should have been called "algebra with respect to the origin" instead!