21. Show that the set of polynomials\[\{c_1, c_2(2x - 1), c_3(6x^2 - 6x + 1)\}\]in \( P(\mathbb{R}) \) is orthogonal with respect to the inner product \(\int_0^1 f(x) \, g(x) \, dx\) for any choice of the constants \( c_1, c_2, c_3 \). For what values of \( c_1, c_2, c_3 \) is the set orthonormal?

Answered: Image /qna-images/answer/b93fb514-767b-4577-87c5-116c21fc8358.jpg

Answered: 21. Show that the set of polynomials…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

1. Let V = P₂(x) and define = 2ƒ(0)g(0) + f(1)g(1) +3ƒ(2)g(2). (a) Prove that this is an inner product (You may use the fact that a quadratic poly- nomial with three roots must be zero). (b) Let f(x) = 1 + x and g(x) orthogonal to f. = x² - 3. Compute Projƒg and the projection of g
f(x)=x²-3 is the minimal polynomial of √3 over Q,
Find the splitting field for f (x) = (x2 + x + 2)(x2 + 2x + 2) overZ3[x]. Write f (x) as a product of linear factors.
6. Determine whether each of these functions is a bijection from R to R. a) f (x) = x² + 1 b) f (x) = x³
part D
2. (a) Show that the polynomials p, (x)=1+x, p,(x)=1+2.x – x², p;(x)=x+x² are linearly independent. (b) Decide if polynomial q(x)=-1+x is in the span{p, (x), P2 (x), P;(x)} .
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p||, ||9||, and d(p, q) for the polynomials in P2. p(x) = 4x + 3x2, g(x) = x - x² (a) (p, q) (b) ||P|| (c) |||| d(p, q) (d)

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with respect to the inner product f f(x) g(x) dx for any choice of the constants C₁, C2, C3. For what values of C₁, C2, C3 is the set orthonormal?