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

[blue2004STi]

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.

 

[ZioX]

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.

 

[blue2004STi]

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.

 

[blue2004STi]

Do you mean by bilinearity, symmetry, and positivity?