A LINEAR TRANSFORMATION $T : V \to W$ satisfies
A$T$ is additive AND homogeneous
B$T(uv) = T(u)T(v)$, multiplicative
C$T(v) = v$, identity always
D$T(\alpha) = T(v)$, scalar-free
Answer & Solution
Correct answer: A. $T$ is additive AND homogeneous
1. A function $T : V \to W$ between vector spaces is LINEAR iff it preserves vector addition and scalar multiplication.
2. Formally: (i) $T(u + v) = T(u) + T(v)$ — ADDITIVITY; (ii) $T(\alpha v) = \alpha T(v)$ — HOMOGENEITY.
3. Equivalently: $T(\alpha u + \beta v) = \alpha T(u) + \beta T(v)$.
4. Consequence: $T(0) = 0$.
5. Option B is multiplicative (not linear). Option C is the identity. Option D is meaningless.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §3.1 (Linear transformations)._
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