Problem 2. In this problem, we will show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map. (a) Give an example of a function T: R² → R such that T(av) = aT (v) for all a ER and v € R², but T is not linear. (b) Give an example of a function S: C→C such that S(z+w) = S(z) + S(w) for all z, w € C, but S is not linear. Here, C is thought of as a complex vector space.
Problem 2. In this problem, we will show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map. (a) Give an example of a function T: R² → R such that T(av) = aT (v) for all a ER and v € R², but T is not linear. (b) Give an example of a function S: C→C such that S(z+w) = S(z) + S(w) for all z, w € C, but S is not linear. Here, C is thought of as a complex vector space.
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 2.** In this problem, we will show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.
(a) Give an example of a function \( T : \mathbb{R}^2 \rightarrow \mathbb{R} \) such that
\[ T(av) = aT(v) \]
for all \( a \in \mathbb{R} \) and \( v \in \mathbb{R}^2 \), but \( T \) is not linear.
(b) Give an example of a function \( S : \mathbb{C} \rightarrow \mathbb{C} \) such that
\[ S(z + w) = S(z) + S(w) \]
for all \( z, w \in \mathbb{C} \), but \( S \) is not linear. Here, \( \mathbb{C} \) is thought of as a complex vector space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac744b86-fb77-4dc8-9b17-1f74c21e67b7%2Fe9f74fbe-b0c8-4e94-b396-de91ea521f11%2Fobzcvv4p_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2.** In this problem, we will show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.
(a) Give an example of a function \( T : \mathbb{R}^2 \rightarrow \mathbb{R} \) such that
\[ T(av) = aT(v) \]
for all \( a \in \mathbb{R} \) and \( v \in \mathbb{R}^2 \), but \( T \) is not linear.
(b) Give an example of a function \( S : \mathbb{C} \rightarrow \mathbb{C} \) such that
\[ S(z + w) = S(z) + S(w) \]
for all \( z, w \in \mathbb{C} \), but \( S \) is not linear. Here, \( \mathbb{C} \) is thought of as a complex vector space.
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