What is a linear map?

A linear map is a transformation that respects the two operations of a vector space. It is the only class of functions linear algebra studies — and that is already a lot: rotations, projections, differentiation, the Fourier transform, the layers of a neural network.

The definition

A map f from a space E to a space F is linear if:

f(u + v) = f(u) + f(v)              it respects addition
f(λ·u)   = λ·f(u)                   it respects scalar multiplication

Both boil down to a single condition, the one checked in practice:

f(λu + μv) = λ·f(u) + μ·f(v)

Plainly: combining then transforming gives the same result as transforming then combining. The order of operations does not matter, and it is this commuting that makes everything computable.

      u, v  ---- combine ----->  λu + μv
        |                            |
     transform                   transform
        |                            |
        v                            v
    f(u), f(v) -- combine ---> λf(u) + μf(v)      <- both paths agree

An immediate consequence

A linear map always sends the zero vector to the zero vector:

f(0) = f(0 · u) = 0 · f(u) = 0

This is the fastest test: if f(0) ≠ 0, the map is not linear.

What is linear, what is not

LINEAR                                  NOT LINEAR
------------------------------------    ---------------------------------
rotation about the origin               translation (f(0) = vector ≠ 0)
projection onto a line/plane            x -> x²  (since (2x)² = 4x², not 2x²)
scaling  x -> 3x                        x -> x + 1  ("affine", not linear)
reflection in a line                    x -> sin(x)
differentiation P -> P'                 x -> |x|
integration f -> integral of f

Beware the classic trap: x -> 2x + 1 is called an affine function, and its graph is indeed a line — but it is not linear in the sense of linear algebra, since f(0) = 1 ≠ 0.

The geometric case

In the plane, linear maps are exactly the transformations that fix the origin and preserve alignment:

    before                       after a projection onto the x-axis

     ^                                ^
     |  •  •                          |
     |    •                           |
  ---+--------->                   ---+--•--•--•--->
     |                                |
     O                                O

the origin stays the origin, a line stays a line (or a point)

A linear map turns a regular grid into another regular grid: the cells may be stretched, sheared or flattened — never curved.

An example outside geometry

On the space of polynomials, differentiation D : P -> P' is linear:

(P + Q)' = P' + Q'          and        (λP)' = λP'

This is what allows linear differential equations to be handled with matrix tools: kernel, image, rank and eigenvalues apply as they stand.

Summary

• f is linear if f(λu + μv) = λf(u) + μf(v).
• Consequence: f(0) = 0 — an immediate test for ruling candidates out.
• Linear: rotations, projections, reflections, scalings, differentiation, integration.
• Not linear: translations, x -> x², x -> x + 1 (affine), x -> |x|.
• A linear map preserves lines and the origin; it never curves anything.