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CSS Pure Mathematics QUESTION #4117
Question 1
For $T: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $T(x, y, z) = (|x|, y + z)$, why is $T$ NOT linear?
  • Fails additivity: $T(u + v) \neq T(u) + T(v)$
  • Fails homogeneity: $T(\lambda v) \neq \lambda T(v)$ for $\lambda < 0$
  • Not defined at origin
  • Both (A) and (B)✔️
Correct Answer Explanation
$T(-1, 0, 0) = (1, 0) \neq -(1, 0) = -T(1, 0, 0)$. Also $T(-1, 0, 0) + T(1, 0, 0) = (2, 0) \neq (0, 0) = T(0, 0, 0)$. Both properties fail.