- {x|f, 7(x) dx = 17. V = C[a, b], S = 0 0
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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Question
Is S closed under addition, scalar multiplication, and is S a subspace of V?
(number 17 on picture)
![Here is the transcription of the text suitable for an educational website:
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11. \( V = M_{33}, \, S = \{ A \mid A \text{ is invertible} \} \)
12. \( V = M_{33}, \, S = \{ A \mid a_{11} + a_{22} + a_{33} = 0 \} \)
13. \( V = M_{44}, \, S = \{ A \mid a_{11} + a_{22} + a_{33} + a_{44} = 0 \} \)
14. \( V = M_{22}, \, S = \{ A \mid A \text{ is singular} \} \)
15. \( V = C[a, b], \, S = \{ f \mid f(a) = 0 \} \)
16. \( V = C[a, b], \, S = \{ f \mid f(a) = 1 \} \)
17. \( V = C[a, b], \, S = \left\{ f \mid \int_{a}^{b} f(x) \, dx = 0 \right\} \)
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**Explanation:**
These are mathematical representations used to define sets \( S \) within different vector spaces \( V \). The elements in these sets satisfy specific conditions:
- **11.** The set \( S \) consists of all invertible matrices \( A \) within the space \( M_{33} \) (3x3 matrices).
- **12.** The set \( S \) includes all 3x3 matrices \( A \) where the sum of the diagonal elements is zero.
- **13.** Similar to 12, but for 4x4 matrices within the space \( M_{44} \).
- **14.** The set \( S \) consists of all singular matrices (non-invertible) within the space \( M_{22} \) (2x2 matrices).
- **15.** The set \( S \) contains all functions in the continuous function space \( C[a, b] \) such that the function evaluates to zero at \( a \).
- **16.** Similar to 15, but the function evaluates to one at \( a \).
- **17.** The set \( S \) consists of all functions](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92dd192b-2b85-4e23-91c3-a8e23aec13bb%2Fe1f4d60e-d33f-49d5-8c3f-84987725bb4c%2Fxjehy55_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the text suitable for an educational website:
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11. \( V = M_{33}, \, S = \{ A \mid A \text{ is invertible} \} \)
12. \( V = M_{33}, \, S = \{ A \mid a_{11} + a_{22} + a_{33} = 0 \} \)
13. \( V = M_{44}, \, S = \{ A \mid a_{11} + a_{22} + a_{33} + a_{44} = 0 \} \)
14. \( V = M_{22}, \, S = \{ A \mid A \text{ is singular} \} \)
15. \( V = C[a, b], \, S = \{ f \mid f(a) = 0 \} \)
16. \( V = C[a, b], \, S = \{ f \mid f(a) = 1 \} \)
17. \( V = C[a, b], \, S = \left\{ f \mid \int_{a}^{b} f(x) \, dx = 0 \right\} \)
---
**Explanation:**
These are mathematical representations used to define sets \( S \) within different vector spaces \( V \). The elements in these sets satisfy specific conditions:
- **11.** The set \( S \) consists of all invertible matrices \( A \) within the space \( M_{33} \) (3x3 matrices).
- **12.** The set \( S \) includes all 3x3 matrices \( A \) where the sum of the diagonal elements is zero.
- **13.** Similar to 12, but for 4x4 matrices within the space \( M_{44} \).
- **14.** The set \( S \) consists of all singular matrices (non-invertible) within the space \( M_{22} \) (2x2 matrices).
- **15.** The set \( S \) contains all functions in the continuous function space \( C[a, b] \) such that the function evaluates to zero at \( a \).
- **16.** Similar to 15, but the function evaluates to one at \( a \).
- **17.** The set \( S \) consists of all functions
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