V = {ax² + bx +3|a, b ≤ R} ≤ P₂.
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
It is not a span and why?
![**Vector Subspace of Polynomials**
The given text describes a subset of a vector space of polynomials of degree less than or equal to 2. Specifically, it defines a set \( V \) as follows:
\[ V = \{ ax^2 + bx + 3 \mid a, b \in \mathbb{R} \} \subseteq \mathbb{P}_2. \]
### Explanation:
- **\( V \)**: This is a subset of polynomials which have the specific form \( ax^2 + bx + 3 \), where both \( a \) and \( b \) are real numbers (\( a, b \in \mathbb{R} \)).
- **\( \mathbb{P}_2 \)**: Represents the vector space of all polynomials of degree less than or equal to 2. This includes all polynomials of the form \( ax^2 + bx + c \) where \( a, b, c \in \mathbb{R} \).
### Notes:
- The polynomials in set \( V \) all have a constant term equal to 3. Thus, they are polynomials with coefficients that vary for \( ax^2 \) and \( bx \) only.
- The inclusion symbol \( \subseteq \) indicates that \( V \) is a subset of \( \mathbb{P}_2 \), meaning every element of \( V \) is also an element of \( \mathbb{P}_2 \).
This formulation is often used to explore properties of subspaces within a defined vector space, such as closure under addition and scalar multiplication.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc495729c-fc87-4b7b-a937-12ad0926c14e%2F88b7ff4c-fa43-4071-ac55-9c75145d67be%2Fs7tno4p_processed.png&w=3840&q=75)
Transcribed Image Text:**Vector Subspace of Polynomials**
The given text describes a subset of a vector space of polynomials of degree less than or equal to 2. Specifically, it defines a set \( V \) as follows:
\[ V = \{ ax^2 + bx + 3 \mid a, b \in \mathbb{R} \} \subseteq \mathbb{P}_2. \]
### Explanation:
- **\( V \)**: This is a subset of polynomials which have the specific form \( ax^2 + bx + 3 \), where both \( a \) and \( b \) are real numbers (\( a, b \in \mathbb{R} \)).
- **\( \mathbb{P}_2 \)**: Represents the vector space of all polynomials of degree less than or equal to 2. This includes all polynomials of the form \( ax^2 + bx + c \) where \( a, b, c \in \mathbb{R} \).
### Notes:
- The polynomials in set \( V \) all have a constant term equal to 3. Thus, they are polynomials with coefficients that vary for \( ax^2 \) and \( bx \) only.
- The inclusion symbol \( \subseteq \) indicates that \( V \) is a subset of \( \mathbb{P}_2 \), meaning every element of \( V \) is also an element of \( \mathbb{P}_2 \).
This formulation is often used to explore properties of subspaces within a defined vector space, such as closure under addition and scalar multiplication.
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