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 pa(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-Pc} and W = R{Pb + Pc), and let B = {Pa. Pb. Pc} and D = {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 (a) where a: P₂(R) → P₂(R) is the projection on Valong W, i.e. a(v) = v for all VE V and a(w) = 0 for all w€ W.
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 pa(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-Pc} and W = R{Pb + Pc), and let B = {Pa. Pb. Pc} and D = {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 (a) where a: P₂(R) → P₂(R) is the projection on Valong W, i.e. a(v) = v for all VE V and a(w) = 0 for all w€ W.
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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Transcribed Image Text: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} and
D = {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 all
VE V and a(w) = 0 for all w€ W.
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