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Let E be any Euclidean sp The following properties hol ap f: EE is an isometry fof=

Let E be any Euclidean sp The following properties hol ap f: EE is an isometry fof=

Advanced Engineering Mathematics
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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Let E be any Euclidean space of finite dimension n, and let f: E → E be any linear map. The following properties hold: (1) The linear map f: EE is an isometry iff fof*= f* of = id. (2) For every orthonormal basis (e₁,..., en) of E, if the matrix of f is A, then the matrix of f* is the transpose AT of A, and f is an isometry iff A satisfies the identities AAT = ATA=In, where In denotes the identity matrix of order n, iff the columns of A form an orthonor- mal basis of R", iff the rows of A form an orthonormal basis of Rn.
Transcribed Image Text:Let E be any Euclidean space of finite dimension n, and let f: E → E be any linear map. The following properties hold: (1) The linear map f: EE is an isometry iff fof*= f* of = id. (2) For every orthonormal basis (e₁,..., en) of E, if the matrix of f is A, then the matrix of f* is the transpose AT of A, and f is an isometry iff A satisfies the identities AAT = ATA=In, where In denotes the identity matrix of order n, iff the columns of A form an orthonor- mal basis of R", iff the rows of A form an orthonormal basis of Rn.
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