SystemScreenshot capturedTap here to view it.Share=[2 0 1]Let M0 1 2 be the matrix of a linear map T: P₂ (R)→ P₂ (R)1 1 1with respect to the basis B = C = {1, x, x²}. What is T(2+5x - 4x²)?ALT 50% ZODeleteVxDo8

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Answered: [201] Let M0 1 2 be the matrix of a…

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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4) Let 4: R³ → R[t]<2 be the linear map that sends Y(e₁) = t + t² 4(e₂) = 1-t 4(e3) = t Write the matrix of the complexified map yc : C³ → C[t]<2 with respect to the complexified standard basis of both vector spaces.
State with reasons whether each of the following maps T : P2(R) → P2(R) is linear. For those maps that are linear, find their matrix with respect to the basis 1, x, x^2. (a) T(f)(x) = f(x + 1), (b) T(f)(x) = f(x) + 1.
Define T: R² →R² by T(x) = Ax, where A is the matrix defined below. Find a basis B for R² with the property that [T] is diagonal. - 3 -6 A = [*] 8 11 •(-19~-0 B= =
4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
The matrix of the linear operator f: R₁[x] → R₁[x] relative to the basis B = {1-2x, 2+x} is (5). Which of the following equals f(1 - 7x)? (a) 20- 15x (b) 20 + 15x (c) -20 - 15x (d) -20 + 15x
y+2 (O)-() x + (-)-( (b) Consider the linear map T: R³ R³ such that Ty i. State the matrix of T in standard basis B of R³. ii. Determine the matrix of T in the basis C = You may assume without proof that C forms a basis. 2 2 of R³.
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1. (a) Express the polynomial q(t) = 3t² + 5t - 5 as a linear combination of the polynomials P₁ (t) = ² + 2t + 1, P2 (t) = 2t² + 5t + 4, P3(t) = 1² + 3t+ 6. (b) Consider the linear map T: R³ R³ such that T Y Z i. State the matrix of T in standard basis B of R³. ii. Determine the matrix of T in the basis C = = 2 {0)·()·()} - -1 y + z x + z y + x You may assume without proof that C forms a basis. 1 of R³.
Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R² → R²: T(x, y) = (6x + 2y, 3x + y) (63) B =
In the vector space P₂ of polynomials of degree at most two,the second column of the matrix of transition from the basis A=(a₁ = 1; a₁ = x; a=x²) to the basis B=(e₁=2+x;e₂=1-x+ 6x²; e₂=1-x-6x² is: OA. 1 1 9 @ -6 OB. 1 OC., (8) O D.

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[201] Let M0 1 2 be the matrix of a linear map T: P₂ (R)→ P₂ (R) 1 1 with respect to the basis B = C = {1, 2, 22). What is T(2+52-4x²)?