Let V be a finite dimensional vector space, with S : V → V a linear operator, and ⟨·, ·⟩ an inner product on V . Show that the following are equivalent: (1) ∥v − w∥ = ∥S(v) − S(w)∥ for all v, w ∈ V . (i.e. S is distance preserving.) (2) ⟨v, w⟩ = ⟨S(v), S(w)⟩ for all v, w ∈ V . (i.e. S is inner-product preserving.)

Let V be a finite dimensional vector space, with S : V → V a linear operator, and ⟨·, ·⟩ an inner product on V . Show that the following are equivalent: (1) ∥v − w∥ = ∥S(v) − S(w)∥ for all v, w ∈ V . (i.e. S is distance preserving.) (2) ⟨v, w⟩ = ⟨S(v), S(w)⟩ for all v, w ∈ V . (i.e. S is inner-product preserving.)

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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