To show that the first four Hermite polynomials form a basis of P3, what theorem should be used? O A. 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. Let V be a p-dimensional vector space, p21. Any linearly independent set of exactly p elements in V is automatically a basis for V. O D. If a vector space V has a basis B= (b,..., bn). then any set in V containing more than n vectors must be linearly dependent. Write the standard basis of the space P, of polynomials, in order of ascending degree. Šimplify 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.

To show that the first four Hermite polynomials form a basis of P3, what theorem should be used? O A. 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. Let V be a p-dimensional vector space, p21. Any linearly independent set of exactly p elements in V is automatically a basis for V. O D. If a vector space V has a basis B= (b,..., bn). then any set in V containing more than n vectors must be linearly dependent. Write the standard basis of the space P, of polynomials, in order of ascending degree. Šimplify 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.

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

The first four Hermite polynomials are 1, 2t, -2+4t, and - 12t + 81. 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?
O A. 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. Let V be a p-dimensional vector space, p21. Any linearly independent set of exactly p elements in V is automatically a basis for V.
O D. If a vector space V has a basis B = (b1, ..., b,), then any set in V containing more than n vectors must be linearly dependent.
Write the standard basis of the space P, of polynomials, in order of ascending degree.
(Simplify your answers. Type expressions usingtas 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 Pg for 2t is
The coordinate vector in P3 for - 2+ 41 is
OOOO

The coordinate vector in Pa for - 12t + 8t° is
How can these vectors be shown to be linearly independent?
O A. Form a matrix using the vectors as columns and find its inverse.
O B. Form a matrix using the vectors as columns and augment it with a vector b.
OC. Form a matrix using the vectors as columns and determine the number of pivots in the matrix.
O D. Form a matrix using the vectors as columns and determine the number of variables in the equation Ax = 0.
Form a matrix using the four coordinate vectors in P, for the Hermite polynomials. In order from
t to right, use the vectors for 1, 2t, -2+41, and - 12t + 81.
Since there are pivots in this matrix, the columns of this matrix
V a linearly independent set.
What is the dimension of the vector space P3?
Since the dimension of Pa is
V the number of elements in the
V set formed by the given Hermite polynomials, the given set of Hermite polynomials
Va basis for P3

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