Let V=R₂₂ [x] with (the inner product, 1
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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span [(110/47)x^2-1 , (65/47)x-1]
![**Orthogonal Complement in Polynomial Spaces**
Given the polynomial vector space \( V = \mathbb{R}_{(2)}[x] \) with the inner product defined by:
\[ \langle f(x), g(x) \rangle = \int_0^1 f(x) \, \overline{g(x)} \, dx \]
we are tasked with finding a basis for the orthogonal complement to the polynomial \( x^2 + x + 1 \).
### Steps:
1. **Inner Product Definition**:
The inner product of two polynomials \( f(x) \) and \( g(x) \) is given by the integral of their product over the interval \([0, 1]\), with conjugation applied to \( g(x) \).
2. **Orthogonal Complement**:
To find the orthogonal complement, we need to determine all polynomials in \( \mathbb{R}_{(2)}[x] \) that are orthogonal to \( x^2 + x + 1 \). This means finding polynomials \( p(x) \) such that:
\[
\int_0^1 p(x) \cdot (x^2 + x + 1) \, dx = 0
\]
### Conclusion:
After solving the integral equation, we can obtain the required polynomial(s) that form the basis for the orthogonal complement of \( x^2 + x + 1 \) in \( \mathbb{R}_{(2)}[x] \).
This problem is a standard exercise in understanding inner products and orthogonality in polynomial vector spaces, often encountered in Linear Algebra or Functional Analysis courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae19e8bd-5915-473c-8a03-9ac053dffb7e%2F95899966-2343-42a5-89cf-fd39135a75c4%2Fjnxrlei_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Orthogonal Complement in Polynomial Spaces**
Given the polynomial vector space \( V = \mathbb{R}_{(2)}[x] \) with the inner product defined by:
\[ \langle f(x), g(x) \rangle = \int_0^1 f(x) \, \overline{g(x)} \, dx \]
we are tasked with finding a basis for the orthogonal complement to the polynomial \( x^2 + x + 1 \).
### Steps:
1. **Inner Product Definition**:
The inner product of two polynomials \( f(x) \) and \( g(x) \) is given by the integral of their product over the interval \([0, 1]\), with conjugation applied to \( g(x) \).
2. **Orthogonal Complement**:
To find the orthogonal complement, we need to determine all polynomials in \( \mathbb{R}_{(2)}[x] \) that are orthogonal to \( x^2 + x + 1 \). This means finding polynomials \( p(x) \) such that:
\[
\int_0^1 p(x) \cdot (x^2 + x + 1) \, dx = 0
\]
### Conclusion:
After solving the integral equation, we can obtain the required polynomial(s) that form the basis for the orthogonal complement of \( x^2 + x + 1 \) in \( \mathbb{R}_{(2)}[x] \).
This problem is a standard exercise in understanding inner products and orthogonality in polynomial vector spaces, often encountered in Linear Algebra or Functional Analysis courses.
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