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CSS Pure Mathematics QUESTION #1288
Question 1
A linear transformation \(T:\mathbb{R}^3\to\mathbb{R}^2\) must satisfy, for all \(\mathbf{u},\mathbf{v}\in\mathbb{R}^3\) and scalar \(c\):
  • T(\(\mathbf{u}+\mathbf{v}) = T(\mathbf{u})\cdot T(\mathbf{v})\)
  • T(c\(\mathbf{u}) = c^2T(\mathbf{u})\)
  • T(\(\mathbf{u}+\mathbf{v}) = T(\mathbf{u})+T(\mathbf{v})\) and \(T(c\mathbf{u})=cT(\mathbf{u})\)✔️
  • T must be onto
Correct Answer Explanation
A linear transformation satisfies two properties: (1) Additivity: \(T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})\), and (2) Homogeneity: \(T(c\mathbf{u})=cT(\mathbf{u})\). Both must hold simultaneously. Together they imply \(T(c\mathbf{u}+d\mathbf{v})=cT(\mathbf{u})+dT(\mathbf{v})\).