Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p||, ||q||, and d(p, q) for the polynomials in P2. p(x) = 4x + 3x², g(x) = x - x² (a) (p, q) (b) ||P|| (c) |||| (d) d(p, q)
Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p||, ||q||, and d(p, q) for the polynomials in P2. p(x) = 4x + 3x², g(x) = x - x² (a) (p, q) (b) ||P|| (c) |||| (d) d(p, q)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Using Inner Product for Polynomials in \( P_2 \)
In this exercise, we will explore the inner product of polynomials in the space \( P_2 \) by evaluating the given polynomials \( p(x) \) and \( q(x) \). We will use the inner product definition to find \( \langle p, q \rangle \), \( \|p\| \), \( \|q\| \), and \( d(p, q) \).
#### Inner Product Definition
The inner product is defined as:
\[ \langle p, q \rangle = a_0b_0 + a_1b_1 + a_2b_2 \]
Here, \( p(x) \) and \( q(x) \) are polynomials, and \( a_i \) and \( b_i \) are their respective coefficients.
Given:
\[ p(x) = 4 - x + 3x^2 \]
\[ q(x) = x - x^2 \]
#### Steps to Evaluate
1. **Inner Product \( \langle p, q \rangle \):**
\[ \langle p, q \rangle = (4)(0) + (-1)(1) + (3)(-1) \]
2. **Norm of \( p(x) \):**
\[ \|p\| = \sqrt{ \langle p, p \rangle } \]
\[ \langle p, p \rangle = (4)(4) + (-1)(-1) + (3)(3) \]
3. **Norm of \( q(x) \):**
\[ \|q\| = \sqrt{ \langle q, q \rangle } \]
\[ \langle q, q \rangle = (0)(0) + (1)(1) + (-1)(-1) \]
4. **Distance \( d(p, q) \):**
\[ d(p, q) = \sqrt{ \|p\|^2 + \|q\|^2 - 2 \langle p, q \rangle } \]
### Assignment
Fill in the following blanks based on the steps above:
**(a) Inner Product \( \langle p, q \rangle \)**
\[ \boxed{} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F031412a4-ce24-4200-b630-467aaa8d4c51%2Ffd253651-69d3-4620-9523-048897d62d56%2Fjc48oyt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Inner Product for Polynomials in \( P_2 \)
In this exercise, we will explore the inner product of polynomials in the space \( P_2 \) by evaluating the given polynomials \( p(x) \) and \( q(x) \). We will use the inner product definition to find \( \langle p, q \rangle \), \( \|p\| \), \( \|q\| \), and \( d(p, q) \).
#### Inner Product Definition
The inner product is defined as:
\[ \langle p, q \rangle = a_0b_0 + a_1b_1 + a_2b_2 \]
Here, \( p(x) \) and \( q(x) \) are polynomials, and \( a_i \) and \( b_i \) are their respective coefficients.
Given:
\[ p(x) = 4 - x + 3x^2 \]
\[ q(x) = x - x^2 \]
#### Steps to Evaluate
1. **Inner Product \( \langle p, q \rangle \):**
\[ \langle p, q \rangle = (4)(0) + (-1)(1) + (3)(-1) \]
2. **Norm of \( p(x) \):**
\[ \|p\| = \sqrt{ \langle p, p \rangle } \]
\[ \langle p, p \rangle = (4)(4) + (-1)(-1) + (3)(3) \]
3. **Norm of \( q(x) \):**
\[ \|q\| = \sqrt{ \langle q, q \rangle } \]
\[ \langle q, q \rangle = (0)(0) + (1)(1) + (-1)(-1) \]
4. **Distance \( d(p, q) \):**
\[ d(p, q) = \sqrt{ \|p\|^2 + \|q\|^2 - 2 \langle p, q \rangle } \]
### Assignment
Fill in the following blanks based on the steps above:
**(a) Inner Product \( \langle p, q \rangle \)**
\[ \boxed{} \
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