Why is a linear transformation called linear? [duplicate]

$T(av_1 + bv_2) = aT(v_1) + bT(v_2)$

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.

Solution 1:

Yep, in high school we say $f(x) = 3x + 4$ is a linear function, but as you point out, in linear algebra it's not. It's irritating.

The word for $x \mapsto ax + b$ is affine transformation, and this is the term you'll hear elsewhere in higher math.

Solution 2:

https://hsm.stackexchange.com/questions/2490/why-do-we-call-a-linear-mapping-linear-mapping

As explained there, the term linear mapping was coined by Hermann Graßmann. It describes mappings which preserve the linear structure of a space, meaning the way scaling the length of a vector parameterizes a line. If you apply a linear mapping, the image will still be a line.

Now, that's actually true for affine maps as well, so it could be argued that the high school term, using linear to mean functions of the form $\backslash x \mapsto a\cdot x + b$, is actually more meaningful. But alas, sometimes sub-optimal terminology sticks. There has by now been so much written about linear maps meaning functions that fulfill $f(\mu\cdot \mathbf v + \nu\cdot \mathbf w) = \mu\cdot f(\mathbf v) + \nu\cdot f (\mathbf w)$, that it would mostly cause confusion to use it for anything else.

Solution 3:

Expressions of the form $av_1 + bv_2$ are called linear combinations. Linear transformations are the functions sending linear combinations to linear combinations (preserving coefficients). That is, a function is called linear when it preserves linear combinations.