V = {ax³ + ax² + bx − 2c|a + c = 0, b ≤ R} ≤ P3.
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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Write the sets as a span of the minimum number of vectors. If this is not a span and why?
![The expression defines a set \( V \) of cubic polynomials in the form:
\[ V = \{ ax^3 + ax^2 + bx - 2c \mid a + c = 0, \, b \in \mathbb{R} \} \subseteq \mathcal{P}_3. \]
### Explanation:
- **Polynomial Description**: Each polynomial in the set \( V \) is of the form \( ax^3 + ax^2 + bx - 2c \).
- **Constraints**: The coefficients satisfy the condition \( a + c = 0 \), which implies that \( c = -a \).
- **Variable \( b \)**: \( b \) is any real number, \( b \in \mathbb{R} \).
- **Subset Notation**: \( V \) is a subset of \( \mathcal{P}_3 \), the space of all polynomials of degree up to 3.
This definition outlines specific conditions under which a polynomial belongs to the set \( V \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc495729c-fc87-4b7b-a937-12ad0926c14e%2F1d61593c-1548-4ee2-85f2-b01f14cf37b9%2Fgx6q1sp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The expression defines a set \( V \) of cubic polynomials in the form:
\[ V = \{ ax^3 + ax^2 + bx - 2c \mid a + c = 0, \, b \in \mathbb{R} \} \subseteq \mathcal{P}_3. \]
### Explanation:
- **Polynomial Description**: Each polynomial in the set \( V \) is of the form \( ax^3 + ax^2 + bx - 2c \).
- **Constraints**: The coefficients satisfy the condition \( a + c = 0 \), which implies that \( c = -a \).
- **Variable \( b \)**: \( b \) is any real number, \( b \in \mathbb{R} \).
- **Subset Notation**: \( V \) is a subset of \( \mathcal{P}_3 \), the space of all polynomials of degree up to 3.
This definition outlines specific conditions under which a polynomial belongs to the set \( V \).
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