What must a linear map preserve?

Source scope: §3.2. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Move combinations together

The map $T(x,y)=(x+y,2y)$ sends $(2,1)$ to $(3,2)$. It sends $a(1,0)+b(0,1)$ to $a(1,0)+b(1,2)$. The coefficients survive the transformation.

A linear map $T:V\to W$ satisfies, for all inputs and real scalars,

The source omits unnecessary parentheses, writing $Tx$. It writes composition as $ST=S\circ T$: apply $T$ first. A map $V\to V$ is a linear operator. A translation $F(x)=x+b$ with $b\ne0$ is not linear, because $F(0)\ne0$.

Same structure, different objects

A linear map is a vector-space homomorphism. An invertible homomorphism whose inverse is linear is an isomorphism; write $V\cong W$. This means their vector-space structure agrees, not that their elements are literally identical.

The source gives the following construction. Suppose real spaces $V,W$ have dimension $n$, with bases $v_1,\ldots,v_n$ and $w_1,\ldots,w_n$. Define

Each input has unique basis coordinates, so this is well-defined. Adding inputs adds their coefficients; scaling scales them. Therefore $\phi$ is linear. Reversing the basis correspondence gives a linear inverse. Hence $V\cong W$; in particular every real $n$-dimensional space is isomorphic to $\mathbb R^n$. Source footnote 2: The two spaces must use the same field.

For example, polynomials of degree at most one correspond to pairs: $a+bt\leftrightarrow(a,b)$. Adding polynomials and adding pairs produce corresponding coefficients.

Transfer

Does preserving zero prove linearity?