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Let V be a vector space. Prove that a) The zero transformation T(v) = 0 for all ve V is a linear transformation. b) The identity transformation T(v) = v for all v EV is a linear transformation.

Let V be a vector space. Prove that a) The zero transformation T(v) = 0 for all ve V is a linear transformation. b) The identity transformation T(v) = v for all v EV is a linear transformation.

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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## Problem Statement **4.** Let \( V \) be a vector space. Prove that: a) The zero transformation \( T(v) = 0 \) for all \( v \in V \) is a linear transformation. b) The identity transformation \( T(v) = v \) for all \( v \in V \) is a linear transformation. ### Explanation This exercise involves proving two fundamental properties in linear algebra concerning transformations on vector spaces. A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication: 1. **Zero Transformation:** - Here, the transformation \( T: V \to V \) maps every vector \( v \) to the zero vector. - To prove it is a linear transformation, we need to verify: - \( T(u + v) = T(u) + T(v) \) - \( T(cv) = cT(v) \) - Since \( T(v) = 0 \) for all vectors, both of these conditions trivially hold. 2. **Identity Transformation:** - The identity transformation \( T: V \to V \) maps every vector \( v \) to itself. - We need to verify the same properties: - \( T(u + v) = T(u) + T(v) \) implies that \( (u + v) = u + v \). - \( T(cv) = cT(v) \) implies \( cv = cv \). - Both conditions are inherently satisfied, thus confirming linearity. The proof of these properties involves verifying that the transformations maintain the essential operations of vector addition and scalar multiplication.
Transcribed Image Text:## Problem Statement **4.** Let \( V \) be a vector space. Prove that: a) The zero transformation \( T(v) = 0 \) for all \( v \in V \) is a linear transformation. b) The identity transformation \( T(v) = v \) for all \( v \in V \) is a linear transformation. ### Explanation This exercise involves proving two fundamental properties in linear algebra concerning transformations on vector spaces. A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication: 1. **Zero Transformation:** - Here, the transformation \( T: V \to V \) maps every vector \( v \) to the zero vector. - To prove it is a linear transformation, we need to verify: - \( T(u + v) = T(u) + T(v) \) - \( T(cv) = cT(v) \) - Since \( T(v) = 0 \) for all vectors, both of these conditions trivially hold. 2. **Identity Transformation:** - The identity transformation \( T: V \to V \) maps every vector \( v \) to itself. - We need to verify the same properties: - \( T(u + v) = T(u) + T(v) \) implies that \( (u + v) = u + v \). - \( T(cv) = cT(v) \) implies \( cv = cv \). - Both conditions are inherently satisfied, thus confirming linearity. The proof of these properties involves verifying that the transformations maintain the essential operations of vector addition and scalar multiplication.
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