Let P3(R) be the vector space of all real polynomials of at most degrees 3 with the innerproduct defined as(f,g) = [ * f(x)g(x)dx.(a) Find an orthonormal basis B for P3(b) Consider the linear operator T : P3 → P3 defined as T(p) = p". Find the charac-teristic polynomial and the minimal polynomial for T.(c) Find the matrix of T from part (b) with respect to B from part (a).

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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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Let P3(R) be the vector space of all real polynomials of at most degrees 3 with the inner
product defined as
(f,g) = [ * f(x)g(x)dx.
(a) Find an orthonormal basis B for P3
(b) Consider the linear operator T : P3 → P3 defined as T(p) = p". Find the charac-
teristic polynomial and the minimal polynomial for T.
(c) Find the matrix of T from part (b) with respect to B from part (a).

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Could you show me part (c) using the orthogonal(not orthonormal) basis in (a)?

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Could you explain how to find minimal polynomial more in detail? I don't understand why when k=1, [T] = 0 and it is contradicting what(?) and when k=0, why m(x)=x^2 and how does it make [T]^2=0? Isn't [T] is our matrix? If so, what is the relationship between the matrix and the minimal polynomial?

I would appreciate your kind explanation.

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