1)LetT(x, y, z) := (2x + y, x +y + z, y – 32) de R³ in R³be a linear operator. Considering the usual internal product in R³: a)Show that T is a self-adjoint operator but is not orthogonal. b)IF v = (2, –1, 5)andw = (3,0, 1), Make sure that (Tv, w) = (v,Tw).

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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1)LetT(x, y, z) := (2x + y, x + y +
z, y – 32) de R³ in R3be a linear operator.
Considering the usual internal product in R3:
a)Show that T is a self-adjoint operator but is not orthogonal.
b}lf V = (2, –1,5)andw = (3,0, 1), ·
Make sure that (Tv, w) = (v,T'w).
c) Display a base of eigenvectors and eigenvalues ofT and verify that it is an
orthogonal base. From this base, find an orthonormal base.
Transcribed Image Text:1)LetT(x, y, z) := (2x + y, x + y + z, y – 32) de R³ in R3be a linear operator. Considering the usual internal product in R3: a)Show that T is a self-adjoint operator but is not orthogonal. b}lf V = (2, –1,5)andw = (3,0, 1), · Make sure that (Tv, w) = (v,T'w). c) Display a base of eigenvectors and eigenvalues ofT and verify that it is an orthogonal base. From this base, find an orthonormal base.
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