Let T be a one-to-one and onto linear map of II to II. Suppose that (Тx — Ту,Тx — Ту) %3D (х — у,х — у) Vx,у € П. - Prove T is orthogonal. That is show a non-singular linear map which preserves distances is an orthogonal transformation.
Let T be a one-to-one and onto linear map of II to II. Suppose that (Тx — Ту,Тx — Ту) %3D (х — у,х — у) Vx,у € П. - Prove T is orthogonal. That is show a non-singular linear map which preserves distances is an orthogonal transformation.
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 \( T \) be a one-to-one and onto linear map of \(\Pi\) to \(\Pi\). Suppose that
\[
(Tx - Ty, Tx - Ty) = (x - y, x - y) \quad \forall x, y \in \Pi.
\]
Prove \( T \) is orthogonal. That is, show a non-singular linear map which preserves distances is an orthogonal transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fc86557e8-5876-43e1-9140-32f3592c854e%2Ffbbhs7s_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( T \) be a one-to-one and onto linear map of \(\Pi\) to \(\Pi\). Suppose that
\[
(Tx - Ty, Tx - Ty) = (x - y, x - y) \quad \forall x, y \in \Pi.
\]
Prove \( T \) is orthogonal. That is, show a non-singular linear map which preserves distances is an orthogonal transformation.
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