The first four Laguerre polynomials are 1, 1-t, 2-4t+t2, and 6-18t+912-1³. Show that these polynomials form a basis of P3.To show that these polynomials form a basis of P3, what theorem should be used?A. 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.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.b), then any set in V containing more than n vectors must be linearly dependent.D. If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.(b₁....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 polynomials as coordinate vectors relative to the standard polynomial basis of PThe coordinate vector in P3 for 1 isThe coordinate vector in P3 for 1-t isThe coordinate vector in P3 for 2-4t+1² isThe coordinate vector in P3 for 6-18t+ 912-13 ishown to be linearly independent? +91²-1³ is.How can these vectors be shown to be linearly independent?A. Form a matrix using the vectors as columns and find its inverse.B. Form a matrix using the vectors as columns and solve the equation Ax=0 using this matrix as A.C. Form a matrix using the vectors as columns and determine the number of pivots in the matrix.OD. Form a matrix using the vectors as columns and augment it with a vector b.Form a matrix using the four coordinate vectors in P, for the Laguerre polynomials. In order from left to right, use the vectors for 1, 1-1, 2-4t+, and 6-182 +8²-²Since there are pivots in this matrix, the columns of this matrixWhat is the dimension of the vector space P3?Since the dimension of P3 isthe number of elements in thea linearly independent setset formed by the given polynomials, the given set of polynomialsa basis for P

Here, the given vectors are 1,1-t, 2-4t+t2, 6-18t+9t2-t3

Answered: The first four Laguerre polynomials are…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

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…
The set B = {1– 2t + t²,3 – 5t + 4t2 , 2t + 3t²} is a basis of P2. Suppose C = {c1, c2, C3} is another basis of P2 such 4 11 that P = C+B Determine the polynomials c1(t), c2(t), and c3(t). 1 3 -1
B = {3 – 3x², 9 – 2x – 9x², 13 – 4x – 12x?} is a basis for P2. The coordinate vector of p(x) = 4 + 2x – 6x² relative to the basis B is [p]B =
Show that B = {1,1 – x, 2 – 4x + x², 6 – 18x + 9x² – x³} is a basis for P3(R). Hint: Recall that dim(P3(R)) = 4 and note that B has 4 vectors.
Consider the following two ordered bases of R²: {(1, –1), (–2, 3)}, {{-1, –2), (1,3)}. B C a. Find the change of basis matrix from the basis B to the basis C. [id b. Find the change of basis matrix from the basis C to the basis B. [id; ||
Consider the following "standard" basis of R³ over R: --000 = Let q be the real quadratic form (from R³ to R) defined via: q(x1, x2, x3) = 9x² + 7x² + 2x3 + 18x1x2 + 18x1x3 + 10x2x3 Determine [q], the symmetric matrix representing the quadratic form q in terms of the basis &:
Let P2 be the vector space of polynomials of degree 2 or less. Consider the following two ordered bases of P2: {2 + x + x?, - 2 – x2, 5+ x + 2x2}, {-2+х+2?, — 2+ 2а + a?, 1 — 2х — г?}. B C a. Find the change of basis matrix from the basis B to the basis C. [id b. Find the change of basis matrix from the basis C to the basis B. (id] =
Let B be the standard basis of the space P₂ of polynomials. Use coordinate vectors to test whether the following set of polynomials span P2. Justify your conclusion. - 5t+t², 1+5t-t²2, 2+t+t², +8t-3t² G Does the set of polynomials span P₂? O A. Yes; since the matrix whose columns are the B-coordinate vectors of each polynomial has a pivot position in each row, the set of coordinate vectors spans R². By isomorphism between R² and P2, the set of polynomials spans P2. O B. No; since the matrix whose columns are the B-coordinate vectors of each polynomial does not have a pivot position in each row, the set of coordinate vectors does not span R³. By isomorphism between R³ and P2, the set of polynomials does not span P2. OC. No; since the matrix whose columns are the B-coordinate vectors of each polynomial does not have a pivot position in each row, the set of coordinate vectors does not span R². By isomorphism between R² and P2, the set of polynomials does not span P2. ⒸD. Yes; since the…
Let A = 0 -2 0 2 -3 -2 2 1 -4 2 -2 1 6 4 -3 -2 4 -2 2 a. A basis for the column space of A is { }. You should be able to explain and justify your answer. Enter a coordinate vector, such as or , or a comma separated list of coordinate vectors, such as , or ,. b. The dimension of the column space of A is answer): A. rref(A) has a pivot in every row. B. The basis we found for the column space of A has three vectors. OC. Three of the five columns in rref(A) have pivots. OD. Two of the five columns in rref(A) have pivots. □ E. rref(A) has a pivot in every column. □ F. rref(A) is the identity matrix. c. The column space of A is a subspace of d. The geometry of the column space of A is the origin inside R^5 because (select all correct answers -- there may be more than one correct because each column vector in A is a vector in R^4
Consider the basis {1, 1 - t, 1 + t + t2} of the space P2 of polynomials of degree at most two. Find the coordinate vector of t - t2 with respect to the standard basis.

Recommended textbooks for you

Algebra and Trigonometry (6th Edition)

Algebra

ISBN:

9780134463216

Author:

Robert F. Blitzer

Publisher:

PEARSON

Contemporary Abstract Algebra

Algebra

ISBN:

9781305657960

Author:

Joseph Gallian

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra and Trigonometry (6th Edition)

Algebra

ISBN:

9780134463216

Author:

Robert F. Blitzer

Publisher:

PEARSON

Contemporary Abstract Algebra

Algebra

ISBN:

9781305657960

Author:

Joseph Gallian

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra And Trigonometry (11th Edition)

Algebra

ISBN:

9780135163078

Author:

Michael Sullivan

Publisher:

PEARSON

Introduction to Linear Algebra, Fifth Edition

Algebra

ISBN:

9780980232776

Author:

Gilbert Strang

Publisher:

Wellesley-Cambridge Press

College Algebra (Collegiate Math)

Algebra

ISBN:

9780077836344

Author:

Julie Miller, Donna Gerken

Publisher:

McGraw-Hill Education

SEE MORE TEXTBOOKS