2. (a) Show that the list {p o(x), p 1(x), p 2(x),...,p n(x),...,} wherePo(x) = 1, P₁(x) = 1 + x,pj(x) = 1 + x + ·+x²,is a basis of the vector space F[x] of all polynomials with coeficients in F.(b) What are the coordinates of the vector xn relative to the basis (po(x), p₁(x),..., pj(x), ...)?

2. (a) Given list is p0x, p1x, p2x, ..., pnx, ..., where p0x=1, p1x=1+x, ......,…

Answered: 2. (a) Show that the list {p o(x), p…

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

Let W = {(x1, x2, x3) € R³ : x₁ + 2x2 3x3 = 0} (a) Find a basis of W. (b) Give an example of a subspace L in R³ such that R³ = WL. Justify your statement.
Find a basis {p(x), g(x)} for the vector space {f(x) = P₂(1) | ƒ'(−2) = f(1)} where P₂(x) is the vector space of polynomials in with degree less than or equal to 2. |p(x) = , q(x) =
Find a basis {p(r), q(x)} for the vector space {f(x) e P2[z] | f' (-8) = f(1)} where P2[T] is the vector space of polynomials in x with degree at most 2.
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].
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.
Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤ b is a vector space, with addition and numerical multiplication defined by (f+g)(x) = f(x) + g(x) and (tf)(x) = tf(x).
Let fi = 1+2x + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao +a1x+azx²[ao – 2a1 +a2 = 0} of R2[r]. Let g = 1+5x + 9x². If [g]g = (,) then s = a) -7 b) -5 c) 5 d) 7 e) g is not in V.
= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).

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