If f(x) and g(x) are arbitrary polynomials of degree at most 1, then the mapping (f,g) = f(-2)g(-2) + f(2)g(2) defines an inner product in P2. Use this inner product to find (f, g), ||f||, ||g||, and the angle af between f(x) and g(x) for f(x) = 5x +7 and g(x) = -5x. (f,g) = ||f|| = ||g|| = radional
If f(x) and g(x) are arbitrary polynomials of degree at most 1, then the mapping (f,g) = f(-2)g(-2) + f(2)g(2) defines an inner product in P2. Use this inner product to find (f, g), ||f||, ||g||, and the angle af between f(x) and g(x) for f(x) = 5x +7 and g(x) = -5x. (f,g) = ||f|| = ||g|| = radional
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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![### Inner Product and Angles Between Polynomials
Given two arbitrary polynomials \( f(x) \) and \( g(x) \) of degree at most 1, the mapping
\[ \langle f, g \rangle = f(-2)g(-2) + f(2)g(2) \]
defines an inner product in \( P_2 \).
#### Objective
Using this inner product, we aim to find \( \langle f, g \rangle \), \( \|f\| \), \( \|g\| \), and the angle \( \alpha_{f,g} \) between \( f(x) \) and \( g(x) \) for the given polynomials:
\[ f(x) = 5x + 7 \]
\[ g(x) = -5x \]
#### Steps to Solve
1. **Calculate the Inner Product \( \langle f, g \rangle \)**:
\[ \langle f, g \rangle = \]
\[
2. **Calculate the Norm \( \| f \| \)**:
\[ \| f \| = \]
\[
3. **Calculate the Norm \( \| g \| \)**:
\[ \| g \| = \]
\[
4. **Calculate the Angle \( \alpha_{f,g} \) in Radians**:
\[ \alpha_{f,g} = \text{acos}\left( \frac{\langle f, g \rangle}{\| f \| \| g \|} \right) \text{ radians} \]
\[
\]
#### Calculation Summaries
- **Inner Product**: Provides a way to measure the "closeness" of two functions.
- **Norms**: Measure the "length" or "magnitude" of each function.
- **Angle**: Measures the directional difference between the two functions in terms of radians.
By applying the definitions and calculations above, you will obtain the desired metrics for comprehensively understanding the relationship between \( f(x) \) and \( g(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F3d1540a8-ab34-4263-a78b-13bad197a8b3%2Fcdrcxpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Inner Product and Angles Between Polynomials
Given two arbitrary polynomials \( f(x) \) and \( g(x) \) of degree at most 1, the mapping
\[ \langle f, g \rangle = f(-2)g(-2) + f(2)g(2) \]
defines an inner product in \( P_2 \).
#### Objective
Using this inner product, we aim to find \( \langle f, g \rangle \), \( \|f\| \), \( \|g\| \), and the angle \( \alpha_{f,g} \) between \( f(x) \) and \( g(x) \) for the given polynomials:
\[ f(x) = 5x + 7 \]
\[ g(x) = -5x \]
#### Steps to Solve
1. **Calculate the Inner Product \( \langle f, g \rangle \)**:
\[ \langle f, g \rangle = \]
\[
2. **Calculate the Norm \( \| f \| \)**:
\[ \| f \| = \]
\[
3. **Calculate the Norm \( \| g \| \)**:
\[ \| g \| = \]
\[
4. **Calculate the Angle \( \alpha_{f,g} \) in Radians**:
\[ \alpha_{f,g} = \text{acos}\left( \frac{\langle f, g \rangle}{\| f \| \| g \|} \right) \text{ radians} \]
\[
\]
#### Calculation Summaries
- **Inner Product**: Provides a way to measure the "closeness" of two functions.
- **Norms**: Measure the "length" or "magnitude" of each function.
- **Angle**: Measures the directional difference between the two functions in terms of radians.
By applying the definitions and calculations above, you will obtain the desired metrics for comprehensively understanding the relationship between \( f(x) \) and \( g(x) \).
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