Let S : ℝ p → ℝ n and T : ℝ n → ℝ m be linear transformations. Show that the mapping x ↦ T ( S ( x )) is a linear transformation (from ℝ p to ℝ m ). [ Hint: Compute T ( S ( c u + d v )) for u , v in ℝ p and scalars c and d . Justify each step of the computation, and explain why this computation gives the desired conclusion.]
Let S : ℝ p → ℝ n and T : ℝ n → ℝ m be linear transformations. Show that the mapping x ↦ T ( S ( x )) is a linear transformation (from ℝ p to ℝ m ). [ Hint: Compute T ( S ( c u + d v )) for u , v in ℝ p and scalars c and d . Justify each step of the computation, and explain why this computation gives the desired conclusion.]
Solution Summary: The author explains that the mapping xmapsto Tleft is a linear transformation.
Let S : ℝp → ℝn and T : ℝn → ℝm be linear transformations. Show that the mapping x ↦ T(S(x)) is a linear transformation (from ℝp to ℝm). [Hint: Compute T(S(cu + dv)) for u, v in ℝp and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.]
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