QUESTION 3 (Based on Golan Ch 8: Representation of Linear Transformations by Matrices, p133)Let P₂(R) be the vector space of polynomials of degree at most 2 over R and let p₂(X), Pb(X), Pc(X) €P2(R) be the Lagrange polynomials associated with the distinct real numbers a, b, c respectively.Let V = R{Pa + Pb. Pa + Pb - Pe} and W = R{Pb + Pc), and let B = {Pa: Pb. Pc} andD = {Pa + Pb. Pa + Pb = Pc: Pb + Pc}.(3.1) Show that D is linearly independent.(3.2) Show that P₂(R) = V ⓇW.(3.3) Find the change-of-basis matrix PBD from B to D.(3.4) Find B(a) where a: P₂(R) → P₂(R) is the projection on Valong W, i.e. a(v) = v for allVE V and a(w) = 0 for all w€ W.

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Answered: QUESTION 3 (Based on Golan Ch 8:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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