linear_transformation
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Definition
A function $T: V \to W$ between vector spaces is a linear transformation (or linear map) if for all vectors $u, v \in V$ and scalars $c$:
- Additivity: $T(u + v) = T(u) + T(v)$
- Homogeneity: $T(cv) = cT(v)$
Equivalently, $T(au + bv) = aT(u) + bT(v)$ for all scalars $a, b$ and vectors $u, v$.