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Let V₁, V2, and V3 be vector spaces and T₁ : V₁ → V₂ and T2 : V2 → V3 be linear transforma- tions. Show that T₂ 0 T₁ : V₁ → V3 is a linear transformation by verifying: O (a) If v, w € V₁ are vectors then T₂ 0 T₁(v + w) = T₂ 0 T₁(v) + T₂ ° T₁(w) O O O (b) If v € V₁ is a vector and c E R is a scalar then T₂ 0 T₁(cv) = c(T₂ 0 T₁ (v)).

Let V₁, V2, and V3 be vector spaces and T₁ : V₁ → V₂ and T2 : V2 → V3 be linear transforma- tions. Show that T₂ 0 T₁ : V₁ → V3 is a linear transformation by verifying: O (a) If v, w € V₁ are vectors then T₂ 0 T₁(v + w) = T₂ 0 T₁(v) + T₂ ° T₁(w) O O O (b) If v € V₁ is a vector and c E R is a scalar then T₂ 0 T₁(cv) = c(T₂ 0 T₁ (v)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let V₁, V2, and V3 be vector spaces and T₁ : V₁ → V₂ and T2 : V2 → V3 be linear transforma- tions. Show that T₂ 0 T₁ : V₁ → V3 is a linear transformation by verifying: (a) If v, w € V₁ are vectors then T₂ 0 T₁(v + w) = T₂0 T₁ (v) + T₂0 O T₁(w) (b) If v € V₁ is a vector and c E R is a scalar then T₂ ° T₁ (cv) = c(T₂ ° T₁ (v)). O O
Transcribed Image Text:Let V₁, V2, and V3 be vector spaces and T₁ : V₁ → V₂ and T2 : V2 → V3 be linear transforma- tions. Show that T₂ 0 T₁ : V₁ → V3 is a linear transformation by verifying: (a) If v, w € V₁ are vectors then T₂ 0 T₁(v + w) = T₂0 T₁ (v) + T₂0 O T₁(w) (b) If v € V₁ is a vector and c E R is a scalar then T₂ ° T₁ (cv) = c(T₂ ° T₁ (v)). O O
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