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CSS Pure Mathematics QUESTION #1261
Question 1
The transformation \(T:\mathbb{R}^3\to\mathbb{R}^2\) defined by \(T(x,y,z)=(|x|,\,y+z)\) is:
  • Linear
  • Not linear because it is not onto
  • Not linear because \(T(-x,y,z)\neq -T(x,y,z)\)✔️
  • Linear only for \(x\geq0\)
Correct Answer Explanation
A linear transformation requires \(T(c\mathbf{v})=cT(\mathbf{v})\). Check: \(T(-1,0,0)=(|-1|,0)=(1,0)\) but \(-T(1,0,0)=-(1,0)=(-1,0)\). Since \(T(-\mathbf{v})\neq -T(\mathbf{v})\), \(T\) is NOT linear. The \(|x|\) component violates homogeneity.