True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).
True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).
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 (f): True or False
#### Question:
Consider the expression for a potential inner product:
\[
\langle p, q \rangle = p(1)q(1) + p(2)q(2) + p(3)q(3)
\]
Does this expression define an inner product on \( P_2(\mathbb{R}) \)?
#### Detailed Explanation:
In this problem, we are given a specific formula and are asked to determine whether it qualifies as an inner product when applied to the polynomials \( p \) and \( q \) in \( P_2(\mathbb{R}) \), the vector space of polynomials of degree at most 2 with real coefficients.
To qualify as an inner product, the expression must satisfy the following properties:
1. **Conjugate Symmetry** (if in complex space):
\[
\langle p, q \rangle = \overline{\langle q, p \rangle}
\]
For real numbers, this simplifies to:
\[
\langle p, q \rangle = \langle q, p \rangle
\]
2. **Linearity in the First Argument**:
\[
\langle ap_1 + bp_2, q \rangle = a\langle p_1, q \rangle + b\langle p_2, q \rangle
\]
3. **Positive-Definiteness**:
\[
\langle p, p \rangle \geq 0 \quad \text{and} \quad \langle p, p \rangle = 0 \iff p = 0
\]
Given the expression:
\[
\langle p, q \rangle = p(1)q(1) + p(2)q(2) + p(3)q(3)
\]
we need to examine whether it satisfies these properties for elements of \( P_2(\mathbb{R}) \).
**1. Conjugate Symmetry:**
Since we are dealing with real numbers, conjugate symmetry simplifies to symmetry:
\[
\langle p, q \rangle = \langle q, p \rangle
\]
That is,
\[
p(1)q(1) + p(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58141e62-2e25-4eb0-81c0-e118f3946948%2F7d080297-0bc9-46d2-bead-1abdccb56140%2F8gvwlg.png&w=3840&q=75)
Transcribed Image Text:### Problem (f): True or False
#### Question:
Consider the expression for a potential inner product:
\[
\langle p, q \rangle = p(1)q(1) + p(2)q(2) + p(3)q(3)
\]
Does this expression define an inner product on \( P_2(\mathbb{R}) \)?
#### Detailed Explanation:
In this problem, we are given a specific formula and are asked to determine whether it qualifies as an inner product when applied to the polynomials \( p \) and \( q \) in \( P_2(\mathbb{R}) \), the vector space of polynomials of degree at most 2 with real coefficients.
To qualify as an inner product, the expression must satisfy the following properties:
1. **Conjugate Symmetry** (if in complex space):
\[
\langle p, q \rangle = \overline{\langle q, p \rangle}
\]
For real numbers, this simplifies to:
\[
\langle p, q \rangle = \langle q, p \rangle
\]
2. **Linearity in the First Argument**:
\[
\langle ap_1 + bp_2, q \rangle = a\langle p_1, q \rangle + b\langle p_2, q \rangle
\]
3. **Positive-Definiteness**:
\[
\langle p, p \rangle \geq 0 \quad \text{and} \quad \langle p, p \rangle = 0 \iff p = 0
\]
Given the expression:
\[
\langle p, q \rangle = p(1)q(1) + p(2)q(2) + p(3)q(3)
\]
we need to examine whether it satisfies these properties for elements of \( P_2(\mathbb{R}) \).
**1. Conjugate Symmetry:**
Since we are dealing with real numbers, conjugate symmetry simplifies to symmetry:
\[
\langle p, q \rangle = \langle q, p \rangle
\]
That is,
\[
p(1)q(1) + p(
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