The first four Hermite polynomials are 1, 2t, -2 +41², and 12t+8t³. These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of P3. C.. To show that the first four Hermite polynomials form a basis of P3, what theorem should be used? OA. If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors. O B. Let H be a subspace of a finite-dimensional vector space V. Any linearly independent set in H can be expanded, if necessary, to a basis for H. OC. If a vector space V has a basis B = = {b₁₁ 1... bn, then any set in V containing more than n vectors must be linearly dependent. O D. Let V be a p-dimensional vector space, p≥ 1. Any linearly independent set of exactly p elements in V is automatically a basis for V. Write the standard basis of the space P3 of polynomials, in order of ascending degree. (Simplify your answers. Type expressions using t as the variable. Use a comma to separate answers as needed.) Express each of the Hermite polynomials as coordinate vectors relative to the standard polynomial basis of P3. The coordinate vector in P3 for 1 is.
The first four Hermite polynomials are 1, 2t, -2 +41², and 12t+8t³. These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of P3. C.. To show that the first four Hermite polynomials form a basis of P3, what theorem should be used? OA. If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors. O B. Let H be a subspace of a finite-dimensional vector space V. Any linearly independent set in H can be expanded, if necessary, to a basis for H. OC. If a vector space V has a basis B = = {b₁₁ 1... bn, then any set in V containing more than n vectors must be linearly dependent. O D. Let V be a p-dimensional vector space, p≥ 1. Any linearly independent set of exactly p elements in V is automatically a basis for V. Write the standard basis of the space P3 of polynomials, in order of ascending degree. (Simplify your answers. Type expressions using t as the variable. Use a comma to separate answers as needed.) Express each of the Hermite polynomials as coordinate vectors relative to the standard polynomial basis of P3. The coordinate vector in P3 for 1 is.
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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Transcribed Image Text:The first four Hermite polynomials are 1, 2t, −2+4t², and − 12t+8t³. These polynomials arise naturally in the study of certain important differential equations in
mathematical physics. Show that the first four Hermite polynomials form a basis of P3.
To show that the first four Hermite polynomials form a basis of P3, what theorem should be used?
A. If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.
B. Let H be a subspace of a finite-dimensional vector space V. Any linearly independent set in H can be expanded, if necessary, to a basis for H.
C. If a vector space V has a basis B = {b₁, ..., bn}, then any set in V containing more than n vectors must be linearly dependent.
O D. Let V be a p-dimensional vector space, p≥ 1. Any linearly independent set of exactly p elements in V is automatically a basis for V.
Write the standard basis of the space P3 of polynomials, in order of ascending degree.
(Simplify your answers. Type expressions using t as the variable. Use a comma to separate answers as needed.)
Express each of the Hermite polynomials as coordinate vectors relative to the standard polynomial basis of P3.
The coordinate vector in P3 for 1 is
The coordinate vector in P3 for 2t is
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