Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
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
icon
Related questions
Question
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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS