Linear Algebra hw help

1. Nov 6, 2007

blue2004STi

1. The problem statement, all variables and given/known data
Prove that any basis of R^n is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined?

2. Relevant 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.

3. The attempt at a solution
See above.

2. Nov 6, 2007

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.

Last edited: Nov 6, 2007
3. Nov 6, 2007

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.

4. Nov 6, 2007

blue2004STi

Do you mean by bilinearity, symmetry, and positivity?