Let V = P₂(R) denote the space of real-valued polyomials of degree at most 2, and letS = {x, 1-x, 1 – x²}(a) Show that S is a basis for V.(b) Let T : V → V be given by Tp(x) = xp(x). Find the matrix of T with respectto S.(c) Compute kerT. Is T invertible?

Answered: Image /qna-images/answer/e7116dd9-454f-4c7d-a21e-260c86606278.jpg

Answered: Let V = P₂(R) denote the space of…

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 T: V → V be a linear operator whose characteristic polynomial is pT(x)=x2(x−1)3(x+2)2. Judge each item below as true or false. (A) If the minimal polynomial of T is pT(x) then [T]BB is diagonal for some basis B of V.(B) Let A be the matrix of T with respect to some basis of V. Then, A2(A−I)3(A+2I)2 ≠ 0.
Let f be the bilinear form on R2 defined by f[(x1, x2), (y1, y2)]= 3x1y1-2x1y2+4x2y1-x2y2 then find the matrix A of f in the basis {u1(1, 1)=u2(1, 2)}
9
Consider the basis S = {v1, V2, V3} for R, where v1=(1, 1, 1), v2 = (1, 1, 0), and v3 (1,0,0), and let T:R3 R3 be the linear operator for %3D %3D which T(v1) = (8, - 1, 16), T(v2) = (9,0, 1), T(v3) = (- 1, 17, 1) Find a formula for T(x1, X2, X3), and use that formula to find T(8, 16, - 1). T(8, 16, - 1) = (
Show that if is diagonalizable and T:V→V (Chr(T)) (t) = (₁ t) ... (Ant) for some real numbers then of S A₁,..., An det[T]³ = \₁ · for any ordered basis V • An '= C n II vi i=1
2. For each of the following linear operators T on a vector space V, test T for diagonalisability. If T is diagonal, find a basis 3 for V such that [T]8 is a diagonal matrix. (a) V = P2(R) and T(f(x)) = f'(x) + f"(x). (b) V = P2(R) and T(ax² + bx +c) = cx² + bx + a.
Let A be the matrix defined by [4 1 A = 0 -3 Let T : R? → R² be the map defined by T(x) = Ax. (2, –3). Compute A10(x) by using the matrix (b) M(T) above. Don't forget to convert x to the basis B, and then Let x = convert back.
Define T:R2→R² by T(x) = Ax. Find a basis B for R2 with the property that [T]B is diagonal. 3 A = 1 - 1 A basis for R2 with the property that [T]B is diagonal is (Use a comma to separate answers as needed.)

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

Let V = P₂(R) denote the space of real-valued polyomials of degree at most 2, and let S = {x, 1x, 1 — x²} (a) Show that S is a basis for V. (b) Let T : V → V be given by Tp(x) = xp(x). Find the matrix of T with respect to S. (c) Compute kerT. Is T invertible?