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.)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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