Problem 3. Let V, W be vector spaces over F, and let U be a subspace of V. Suppose SE L(U, W) is such that S 0. Define T: V → W by Prove that T is not linear. T(v) = JSv 10 if v EU, if v EV and v U.
Problem 3. Let V, W be vector spaces over F, and let U be a subspace of V. Suppose SE L(U, W) is such that S 0. Define T: V → W by Prove that T is not linear. T(v) = JSv 10 if v EU, if v EV and v U.
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 3.** Let \( V, W \) be vector spaces over a field \( \mathbb{F} \), and let \( U \) be a subspace of \( V \). Suppose \( S \in \mathcal{L}(U,W) \) is such that \( S \neq 0 \). Define \( T: V \rightarrow W \) by
\[
T(v) =
\begin{cases}
Sv & \text{if } v \in U, \\
0 & \text{if } v \in V \text{ and } v \notin U.
\end{cases}
\]
Prove that \( T \) is not linear.
---
**Problem 4.** Let \( V, W \) be vector spaces over \( \mathbb{F} \), and let \( U \) be a subspace of \( V \). Suppose \( S \in \mathcal{L}(U,W) \). Prove that there exists a linear map \( T \in \mathcal{L}(V,W) \) that extends \( S \), that is,
\[
Tv = Sv \quad \text{for all } v \in U.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac744b86-fb77-4dc8-9b17-1f74c21e67b7%2F11340418-b030-4ced-b9d0-2bebf602bc82%2F6u8uyj6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3.** Let \( V, W \) be vector spaces over a field \( \mathbb{F} \), and let \( U \) be a subspace of \( V \). Suppose \( S \in \mathcal{L}(U,W) \) is such that \( S \neq 0 \). Define \( T: V \rightarrow W \) by
\[
T(v) =
\begin{cases}
Sv & \text{if } v \in U, \\
0 & \text{if } v \in V \text{ and } v \notin U.
\end{cases}
\]
Prove that \( T \) is not linear.
---
**Problem 4.** Let \( V, W \) be vector spaces over \( \mathbb{F} \), and let \( U \) be a subspace of \( V \). Suppose \( S \in \mathcal{L}(U,W) \). Prove that there exists a linear map \( T \in \mathcal{L}(V,W) \) that extends \( S \), that is,
\[
Tv = Sv \quad \text{for all } v \in U.
\]
Transcribed Image Text:In the problems below, F denotes ℝ or ℂ.
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