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Question 4: Let V and W be vector spaces over R, and let T: V → W be a linear map. Suppose 7 € V is a vector such that T(7) + T() = Öw. Prove that 7 € Nul(T).

Question 4: Let V and W be vector spaces over R, and let T: V → W be a linear map. Suppose 7 € V is a vector such that T(7) + T() = Öw. Prove that 7 € Nul(T).

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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Please solve problem 4
### Question 4: Let \( V \) and \( W \) be vector spaces over \( \mathbb{R} \), and let \( T: V \to W \) be a linear map. Suppose \( \vec{v} \in V \) is a vector such that \( T(\vec{v}) + T(\vec{v}) = \vec{0}_W \). Prove that \( \vec{v} \in \text{Nul}(T) \). **Explanation:** - \( V \) and \( W \) are vector spaces over the real numbers \( \mathbb{R} \). - \( T \) is a linear map from vector space \( V \) to vector space \( W \). - \( \vec{v} \in V \) is a vector such that when applied to \( T \), the sum of \( T(\vec{v}) \) with itself equals the zero vector \( \vec{0}_W \) in \( W \). You are required to prove that \( \vec{v} \) lies in the null space of \( T \), denoted \( \text{Nul}(T) \). The null space is the set of all vectors \( \vec{v} \) in \( V \) such that \( T(\vec{v}) = \vec{0}_W \).
Transcribed Image Text:### Question 4: Let \( V \) and \( W \) be vector spaces over \( \mathbb{R} \), and let \( T: V \to W \) be a linear map. Suppose \( \vec{v} \in V \) is a vector such that \( T(\vec{v}) + T(\vec{v}) = \vec{0}_W \). Prove that \( \vec{v} \in \text{Nul}(T) \). **Explanation:** - \( V \) and \( W \) are vector spaces over the real numbers \( \mathbb{R} \). - \( T \) is a linear map from vector space \( V \) to vector space \( W \). - \( \vec{v} \in V \) is a vector such that when applied to \( T \), the sum of \( T(\vec{v}) \) with itself equals the zero vector \( \vec{0}_W \) in \( W \). You are required to prove that \( \vec{v} \) lies in the null space of \( T \), denoted \( \text{Nul}(T) \). The null space is the set of all vectors \( \vec{v} \) in \( V \) such that \( T(\vec{v}) = \vec{0}_W \).
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