**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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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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

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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.

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