linear transformation

A map preserving addition and scalar multiplication: $T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v)$ .

A transformation $T:V\to W$ is linear when it satisfies additivity, $T(\mathbf u+\mathbf v)=T(\mathbf u)+T(\mathbf v)$ , and homogeneity, $T(c\mathbf u)=cT(\mathbf u)$ , for all inputs and scalars — equivalently, $T$ preserves every linear combination: $T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v)$ . Homogeneity with $c=0$ forces $T(\mathbf 0)=\mathbf 0$ for any linear $T$ , the fastest way to disqualify a candidate.