Let B be the basis of P, consisting of the Hermite polynomials 1, 2t, -2+ 4t2, and - 12t + 81°, and let p(t) = 7- 16t2-8t. Find the coordinate vector of p relative to B. [ple =D
Let B be the basis of P, consisting of the Hermite polynomials 1, 2t, -2+ 4t2, and - 12t + 81°, and let p(t) = 7- 16t2-8t. Find the coordinate vector of p relative to B. [ple =D
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 Statement:**
Let \( B \) be the basis of \( P_3 \) consisting of the Hermite polynomials \( 1 \), \( 2t \), \( -2 + 4t^2 \), and \( -12t + 8t^3 \); and let \( p(t) = 7 - 16t^2 - 8t^3 \). Find the coordinate vector of \( p \) relative to \( B \).
\[ [p]_B = \boxed{} \]
**Explanation:**
This problem involves finding the coordinates of the polynomial \( p(t) \) in terms of a given basis \( B \). The basis \( B \) consists of the Hermite polynomials:
1. \( 1 \)
2. \( 2t \)
3. \( -2 + 4t^2 \)
4. \( -12t + 8t^3 \)
The polynomial \( p(t) \) is already provided as:
\[ p(t) = 7 - 16t^2 - 8t^3 \]
**Steps to Solve:**
1. We need to express \( p(t) \) as a linear combination of the basis elements. This can be written as:
\[ p(t) = a_1(1) + a_2(2t) + a_3(-2 + 4t^2) + a_4(-12t + 8t^3) \]
where \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \) are the coefficients we need to determine.
2. Rearrange and combine like terms to compare coefficients of \( t^0 \), \( t^1 \), \( t^2 \), and \( t^3 \) from both sides.
3. Form a system of linear equations based on these coefficients.
4. Solve the system of equations to find \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \).
5. The coordinate vector \( [p]_B \) will then be \([a_1, a_2, a_3, a_4]\).
Insert the steps in detailed mathematical form on an educational website to help students understand the transition step-by-step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2F969e41a9-0242-4441-ae58-d3a8152cb216%2F4b6e6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( B \) be the basis of \( P_3 \) consisting of the Hermite polynomials \( 1 \), \( 2t \), \( -2 + 4t^2 \), and \( -12t + 8t^3 \); and let \( p(t) = 7 - 16t^2 - 8t^3 \). Find the coordinate vector of \( p \) relative to \( B \).
\[ [p]_B = \boxed{} \]
**Explanation:**
This problem involves finding the coordinates of the polynomial \( p(t) \) in terms of a given basis \( B \). The basis \( B \) consists of the Hermite polynomials:
1. \( 1 \)
2. \( 2t \)
3. \( -2 + 4t^2 \)
4. \( -12t + 8t^3 \)
The polynomial \( p(t) \) is already provided as:
\[ p(t) = 7 - 16t^2 - 8t^3 \]
**Steps to Solve:**
1. We need to express \( p(t) \) as a linear combination of the basis elements. This can be written as:
\[ p(t) = a_1(1) + a_2(2t) + a_3(-2 + 4t^2) + a_4(-12t + 8t^3) \]
where \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \) are the coefficients we need to determine.
2. Rearrange and combine like terms to compare coefficients of \( t^0 \), \( t^1 \), \( t^2 \), and \( t^3 \) from both sides.
3. Form a system of linear equations based on these coefficients.
4. Solve the system of equations to find \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \).
5. The coordinate vector \( [p]_B \) will then be \([a_1, a_2, a_3, a_4]\).
Insert the steps in detailed mathematical form on an educational website to help students understand the transition step-by-step.
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