Proving Orthonormal Basis & Uniqueness of Inner Product | Linear Algebra HW Help

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Homework Help Overview

The discussion revolves around proving that any basis of R^n can be considered an orthonormal basis with respect to some inner product, and whether this inner product is uniquely determined. The subject area is linear algebra, focusing on concepts of inner products and orthonormality.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are exploring how to define inner products in R^n and the role of symmetric matrices in this context. There is uncertainty about the uniqueness of the inner product that can make a given basis orthonormal, and participants are questioning the definitions and properties involved, such as bilinearity, symmetry, and positivity.

Discussion Status

The discussion is ongoing with participants seeking clarification on definitions and properties related to inner products. Some have expressed confusion about the original poster's questions, indicating a need for further exploration of the concepts involved.

Contextual Notes

There appears to be some ambiguity regarding the definitions of inner products and the conditions under which they apply, as well as the implications of uniqueness in this context.

blue2004STi
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Homework Statement


Prove that any basis of R^n is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined?


Homework Equations


I am not sure where to begin. Should I just define an arbitrary basis for a arbitrary R^n? I mean I think I understand the question about the inner product being uniquely determined but I am not sure where to begin.


The Attempt at a Solution


See above.
 
Physics news on Phys.org
How do you define inner products in R^n? A familiar question: how and when do symmetric matrices induce inner products on R^n?

Uniquely determined means that there is no other inner product that has those properties. That is, given a basis there is only one inner product that makes the basis an orthonormal set. If you don't know what I meant by symmetric matrices you can just play around with scaling inner products by positive reals.
 
Last edited:
Do you mean in (x^T)Kx or in notation <x,x> or in formula? I'm not going to lie I'm a bit confused with what you're asking.
 
Do you mean by bilinearity, symmetry, and positivity?
 

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