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Given a vector space V, define what it means for a function (, ·) : V × V → V to be an inner product on V. Then, define what it means for a subset S of V to be an orthogonal set with respect to this inner product. Finally, define what it means for a linear transformation T : V –→ V to preserve inner products.

Given a vector space V, define what it means for a function (, ·) : V × V → V to be an inner product on V. Then, define what it means for a subset S of V to be an orthogonal set with respect to this inner product. Finally, define what it means for a linear transformation T : V –→ V to preserve inner products.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section: Chapter Questions
Problem 16RQ
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Please explain and elaborate as much as possible. You can answer is in whatever way you want (ex. paragraph), and feel free to make examples to prove the points. The flow should be logical and concise. I'll leave a like for sure. Thank you. 

 

Given a vector space V, define what it means for a function (, ·) :V × V → V to be an inner product on V. Then, define what it means for a subset S of V to be an orthogonal set with respect to this inner product. Finally, define what it means for a linear transformation T : V → V to preserve inner products.
Transcribed Image Text:Given a vector space V, define what it means for a function (, ·) :V × V → V to be an inner product on V. Then, define what it means for a subset S of V to be an orthogonal set with respect to this inner product. Finally, define what it means for a linear transformation T : V → V to preserve inner products.
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